Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems [Revised] 0821831712, 9780821831717

Table of contents :
Contents
Preface to the Revised English Edition
Foreword to the Second Edition
Introduction
I. Review of Matrices and Quadratic Forms
II. Oscillatory Matrices
III. Small Oscillations of Mechanical Systems with n Degrees of Freedom
IV. Small Oscillations of Mechanical Systems with an Infinite Number of Degrees of Freedom
V. Sign-Definite Matrices
Supplement I. A Method of Approximate Calculation of Eigenvalues and Eigenvectors of an Oscillatory Matrix
Supplement II. On a Remarkable Problem for a String with Beads and Continued Fractions of Stieltjes
Remarks
References
Index

Citation preview

AND

OSCILLATION MATRICES KERNELS AND SMALL VIBRATIONS OF MECHANICAL SYSTEMS REVISED EDITION

F. R. GANTMACHER M. G. KREIN

AMS CHELSEA PUBLISHING American Mathematical Society • Providence, Rhode Island

http://dx.doi.org/10.1090/chel/345

AND

OSCILLATION MATRICES KERNELS AND SMALL VIBRATIONS OF MECHANICAL SYSTEMS REVISED EDITION

F. R. GANTMACHER M. G. KREIN

AMS CHELSEA PUBLISHING American Mathematical Society • Providence, Rhode Island

2000 Mathematics Subject Classification. Primary 34C10, 45A05; Secondary 70Jxx.

Library of Congress Cataloging-in-Publication Data Gantmakher, F. R. (Feliks Ruvimovich) [Ostsilliatsionnye matritsy i iadra i malye kolebaniia mekhanicheskikh sistem. English] Oscillation matrices and kernels and small vibrations of mechanical systems / F. R. Gantmacher, M. G. Krein.—Rev. ed. p. cm. Includes bibliographical references and index. ISBN 0-8218-3171-2 (alk. paper) 1. Matrices. 2. Vector analysis. 3. Oscillations. I. Krein, M. G. (Mark Grigorevich), 1907– II. Title.

QA188 .G3613 2002 512.9434—dc21

2002021449

c 2002 by the American Mathematical Society. All rights reserved.  Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. ∞ The paper used in this book is acid-free and falls within the guidelines  established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

07 06 05 04 03 02

Contents Preface to the Revised English Edition

v

Foreword to the Second Edition

vii

Introduction

1

Chapter I. Review of Matrices and Quadratic Forms 1. Matrices and operations on matrices 2. Sylvester’s identity 3. Eigenvalues and eigenvectors of a matrix 4. Real symmetric matrices 5. Reduction of a quadratic form to the principal axes 6. Reduction of a quadratic form to a sum of squares 7. Positive quadratic forms 8. Hadamard’s inequality 9. Simultaneous reduction of two quadratic forms to sums of squares 10. Minimax properties of eigenvalues of a pencil of forms 11. Reduction of a matrix to a triangular form 12. Polynomials of matrices 13. Associated matrices and the Kronecker theorem

9 9 12 14 21 23 28 34 36 42 50 60 63 64

Chapter II. Oscillatory Matrices 1. Jacobi matrices 2. Oscillatory matrices 3. Examples 4. Perron’s theorem 5. Eigenvalues and eigenvectors of an oscillatory matrix 6. A fundamental determinantal inequality 7. Criterion for a matrix to be oscillatory 8. Properties of the characteristic determinant of an oscillatory matrix 9. Eigenvalues of an oscillatory matrix as functions of its elements

67 67 74 76 83 86 91 97

Chapter III. Small Oscillations of Mechanical Systems with n Degrees of Freedom 1. Equations of small oscillations 2. Oscillations of Sturm systems 3. Second method of setting up the equations of small oscillations of mechanical systems 4. Influence functions iii

105 108 113 113 118 129 131

iv

CONTENTS

5. Chebyshev systems of functions 6. The oscillatory character of the influence function of a segmental continuum 7. Influence function of a string 8. Influence function of a rod 9. Small oscillations of an elastic continuum with n concentrated masses 10. Small oscillations of a segmental continuum 11. Oscillations of a system of concentrated masses placed on a multispan beam Chapter IV. Small Oscillations of Mechanical Systems with an Infinite Number of Degrees of Freedom 1. Principal premises 2. Oscillations of a segmental continuum and oscillatory kernels 3. Oscillatory properties of the vibrations of an everywhere-loaded continuum 4. Vibrations of an arbitrarily loaded continuum 5. Harmonic oscillations of multiply supported rods 6. Oscillatory properties of forced vibrations 7. Oscillations of an elastically supported string 8. Forced oscillations of a string 9. The resolvent of an oscillatory single-pair kernel 10. The Sturm–Liouville equations Chapter V. Sign-Definite Matrices 1. Basic definitions 2. Oscillating systems of vectors 3. Markov systems of vectors 4. Eigenvalues and eigenvectors of sign-definite matrices 5. Approximation of a sign-definite matrix by a strictly sign-definite one

136 142 146 148 157 160 163 167 167 177 180 191 205 210 220 223 225 234 245 245 246 258 263 268

Supplement I. A Method of Approximate Calculation of Eigenvalues and Eigenvectors of an Oscillatory Matrix

275

Supplement II. On a Remarkable Problem for a String with Beads and Continued Fractions of Stieltjes

283

Remarks

299

References

305

Index

309

Preface to the Revised English Edition In preping this publication, the following sources were used: 1. F. R. Gantmaher, M. G. Kre in, Oscillcionnye matricy dra i malye kolebani mehaniqeskih sistem, GITTL, Moskva–Leningrad, 1950; 2. F. R. Gantmakher, M. G. Krein, Oscillation matrices and kernels and small vibrations of mechanical systems, AEC-tr-4481 (physics) Office of technical documentation, Department of Commerce, Washington DC, April 1961; 3. F. R. Gantmacher, M. G. Krein, Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen machanischer Systeme, Academie-Verlag, Berlin, 1960. I tried to preserve the terminology and the unique style of the authors (for example, they never use set-theoretic terminology). The only important exception was made for systematic use of the words “eigenvalue” and “eigenvector” in this translation, instead of “characteristic numbers” and “proper vectors” used in the original. We warn the reader that an eigenvalue of a Fredholm integral equation is not the same as an eigenvalue of the corresponding Fredholm operator, but rather its reciprocal. This terminology is traditional and well established in the theory of integral equations (see, for example, [8]) and it is consistent with physical interpretation of these eigenvalues as squares of frequencies. The modern literature on the subjects originated with this book is enormous, so no attempt was made to compose a complete up-to-date bibliography. Only few modern books and survey papers have been added. This new English edition was made possible by generous support of Purdue University and Humboldt Foundation. David Drasin helped very much with correction of English, and Betty Gick typed the manuscript in TeX. Alex Eremenko, Editor of Translation

v

Foreword to the Second Edition The present edition of this book differs from the first edition, published in 1941, at the beginning of the Great Patriotic War, under the title “Oscillatory Matrices and Small Vibrations of Mechanical Systems” in the following respects: Chapter II, devoted to the theory of oscillatory matrices, has been substantially revised. The treatment of the theory is now made more accessible and purposeful. We retain in Chapter II only the material that has direct relation to oscillatory matrices and is used in the application of these matrices to the theory of oscillations of mechanical systems (Chapters III and IV). In Chapter III, devoted to small oscillations of systems with n degrees of freedom, there has been a substantial revision in the section that explains the mechanical properties that cause the oscillatory nature of the matrix of the influence coefficients of a linear continuum (a string or a rod). In addition, this chapter contains a new section (Sec. 5), in which the properties of Chebyshev systems of functions are explained. These systems of functions are used in Chapters III and IV. Chapter IV is essentially new. The main results of this chapter were contained as Appendix I in the first edition in outline form. Chapter IV is a natural continuation and generalization of Chapters II and III. It treats problems of vibrations of systems with infinite number of degrees of freedom. Whereas the mathematical basis for Chapter III is the theory of oscillatory matrices, the natural mathematical tool of Chapter IV is the theory of loaded integral equations with symmetric oscillatory kernel. This theory in its complete form is presented in this book for the first time. In Chapter V various generalizations and supplements to the algebraic investigations of the preceding chapters are gathered. In the first edition, these results were partially covered in Chapter II, and partially in appendices. Two appendices at the end of this book contain new material which was absent in the old edition. In Appendix I we give a development of the iteration method of approximate calculation of eigenvalues and eigenvectors for a class of oscillatory matrices. Appendix II is devoted to the application of continued fractions to the inverse problems of the theory of oscillations – the construction of a mechanical system with finite number of degrees of freedom from its spectral characteristics. We shall not mention here various less important additions, refinements and corrections in various parts of the book. All that is required to understand the entire material of the book, with the exception of Chapter IV, is that the reader be familiar with the principles of calculus and the theory of determinants. Chapter IV requires familiarity with the theory of linear integral equations (at least with symmetric kernels). vii

viii

FOREWORD TO THE SECOND EDITION

In Chapter V the authors used many valuable comments from the late doctoral candidate of the Academy of Sciences, U.S.S.R., Vitold Lvovich Shmulyan, who was killed in action in the Great Patriotic War. These remarks concerning the first edition of the book were sent by him from the front in August 1942. The authors take this opportunity to pay our last homage to the cherished memory of this talented mathematician and patriot.

http://dx.doi.org/10.1090/chel/345/01

Introduction The theory of small oscillations has always served as a source of various algebraic investigations. It is enough to recall that, starting with the problem of vibration of a string with n beads, Sturm (1828–1836) discovered his well-known theorem of higher algebra and developed the theory of Jacobi forms, which were the algebraic prototype of his research on differential equations. In connection with the theory of small oscillations (and certain problems of geometry), it was proved that the roots of the secular equation of a symmetric matrix are real, and a theory of reduction of a quadratic forms to their principal axes was developed. Many new algebraic facts were discovered in connection with the study of the influence of the change of inertia and stiffness on the frequency of an oscillating system. Investigations of stability of small oscillations lead Routh (1877), A.M. Lyapunov (1895), and Hurwitz (1895) to criteria for the roots of algebraic equations to have negative real parts, etc. Many refined algebraic investigations were performed by Soviet mathematicians under the influence of the well–known method of Academician A.N. Krylov (1931) for expanding the secular determinant, whose roots are the frequencies of the oscillations. The purpose of this book is to introduce the reader to a new circle of algebraic ideas, which, in our opinion, represents a natural mathematical basis for investigation of the so-called oscillatory properties of small harmonic vibrations of linear elastic continua (e.g., transverse vibrations of strings, rods, multiple-span beams, torsional oscillations of shafts, etc.). Let us explain what we have in mind in greater detail. Let us consider small transverse oscillations of a linear elastic continuum (a string or a rod), spread along the x axis from x = a to x = b, and having no fastenings (supports) within (a, b) (a segmental continuum). Then the natural harmonic oscillation of the continuum will be given by the formula y(x, t) = ϕ(x) sin(pt + α). Here y(x, t) is the deflection at the point x at the instant t, ϕ(x) an amplitude function, and p the frequency of oscillation. In the general case, there exists an infinite number of natural oscillations with different frequencies p 0 < p1 < p2 < . . . . If the entire mass of the continuum consists of a finite number of concentrated masses, then the number of these frequencies will be finite. A segmental continuum has the following main oscillatory properties: 1

2

INTRODUCTION

1. All the frequencies p are simple (i.e., the amplitude function of the natural oscillation of a given frequency is uniquely determined up to a constant factor). 2. The natural oscillation with the lowest frequency p0 (the fundamental tone) does not have any nodes (ϕ(x) = 0 when a < x < b). 3. The natural oscillations with frequency pj (the j–th overtone) has exactly j nodes. 4. The nodes of two successive overtones alternate. 5. In the oscillations obtained by superposition of natural oscillations with frequencies pk < pl < · · · < pm , the number of sign changes of the deflection fluctuates with time within the limits from k to m.1 Let us consider first transverse oscillations of a system of n masses, m1 , . . . , mn , concentrated in movable points (a 0 when Δs > 0, and then the whole theory is developed as if we had dσ = ds). We make extensive use in these sections of the Chebyshev systems of functions and Markov sequences of functions [4;30k], which are usually encountered in problems of interpolation and approximation and which were used with such skill by Academician A.A. Markov in his remarkable investigations on limiting values of integrals [38a,b]. In Section 5 we investigate vibrations of multiply-supported rods and clarify one maximal characteristic property of nodes of natural vibrations of a rod. In Section 6 we investigate forced oscillations of a segmental continuum under the influence of a harmonically pulsating concentrated force. The problem reduces to an investigation of the oscillatory properties of the resolvent of an oscillatory kernel. It is interesting to note here that in the proof of the main theorem it becomes necessary to load the integral equation with one additional mass; i.e., to introduce an additional jump in the given distribution function. Thus, it becomes clear here that the introduction of loaded integral equations not only is a natural step, dictated by mechanical applications of these equations, but that it also offers new means for establishing various theorems in the general theory of integral equations (including unloaded ones). In the next three sections, the study of the forced oscillations (i.e., the properties of the resolvent) is made deeper to include the case of continua of the type of a string, the influence function of which is a single-pair oscillatory kernel (e.g., torsional oscillations of shafts, longitudinal vibrations of rods, elastically supported strings, etc.).

INTRODUCTION

7

In Sec. 10 we show that every Sturm–Liouville boundary value problem, after a “shift” of the parameter, is equivalent to an integral equation with oscillatory singlepair kernel; it is shown thereby that the investigations of Sturm himself occupy in our general plan only a very particular and subsidiary position. This circumstance would become even more pronounced, were we to discuss in this book the applications of the theory of integral equations with oscillatory kernels to boundary value problems of higher orders, where oscillation theorems are established for the first time [30g,i,j,k,m; 31]. But the scope of these investigations should, apparently, be the subject of a separate book. In Chapter V we return to problems of matrix algebra. Here we study the general theory of sign-definite matrices of the oscillatory type – a natural generalization of oscillatory matrices. In this theory, a great role is played by vector systems which we call “Markov systems” (the analogue of the Markov sequences of functions from Chapter IV). A study of Markov vector systems is the subject of a separate section, No. 3. At the end of the chapter we give “inverse” theorems, which establish the oscillatory properties of a matrix or kernel from their spectral characteristics. The main text is supplemented by two appendices and remarks. In Appendix I we show that the iteration method of approximate calculation of the eigenvalues, when applied to oscillatory matrices (and some more general classes of matrices), admits substantial development. Successive iterations make it possible to obtain estimates from above and from below for the eigenvalues, not only for the first one but also for successive ones. In Appendix II we solve the problem of determination of masses of n beads and their placement on a string from given frequencies of vibrations of this string with beads. It turns out that if the string is symmetrically secured on both ends, then, under the condition that the beads are placed symmetrically with respect to the center of the string (and symmetric beads have equal masses), the problem always has a unique solution (and can be solved using only rational operations), no matter what the specified 2n different frequencies are. In connection with this problem, we clarify the mechanical meaning of the well-known research of Stieltjes on continued fractions. It turns out that the investigations of Stieltjes can be treated as a study, from a certain point of view, of vibrations of a string on which an infinite sequence of beads is placed, accumulating to one end of the string; with this both the length of the string and the total mass of the beads may be infinite (some details concerning these last cases are not contained in the Appendix). In the remarks at the end of the book, we give references to original papers, which are related to the treated problems.

http://dx.doi.org/10.1090/chel/345/02

CHAPTER I

Review of Matrices and Quadratic Forms 1. Matrices and operations on matrices Let there be given a rectangular matrix (table)    a11 a12 . . . a1n    a22 . . . a2n  a A =  21   .....................    am1 am2 . . . amn

(1)

with m rows and n columns, the elements of which aik (i = 1, 2, . . . , m; k = 1, 2, . . . , n) are complex numbers. We introduce the following notation for the minors of this matrix:      ai1 k1 ai1 k2 . . . ai1 kp  i1 i2 . . . ip a i2 k 2 . . . a i2 k p  a A =  i2 k 1  (p ≤ min(m, n)). k1 k2 . . . kp  .......................    a ip k 1 a ip k 2 . . . a ip k p Henceforth we shall consider for the most part square matrices (m = n). In this case we shall abbreviate the matrix (1) by aik n1 , and its determinant by |A| or |aik |n1 . We shall relate to any two square matrices A = aik n1 and B = bik n1 a matrix C = cik n1 , called their sum (C = A + B), in which cik = aik + bik

(i, k = 1, 2, . . . , n),

and the matrix D = dik n1 , called their product (D = AB), in which the elements dik are obtained as “products” of the i-th row of A by the k-th column of B; i.e., dik = ai1 b1k + ai2 b2k + · · · + ain bnk

(i, k = 1, 2, . . . , n).

In addition, if A = aik n1 , and λ is a number, then the product λA is the matrix λaik n1 . Obviously, we define the difference of two matrices as A − B = aik − bik n1 . We can readily verify the following properties of operations on square matrices of the same order: I. A + B = B + A, (A + B) + C = A + (B + C). II. (AB)C = A(BC), (A + B)C = AC + BC, C(A + B) = CA + CB. We emphasize that, generally speaking, AB = BA. In the case when AB = BA, the matrices A and B are called permutable. 9

10

I. REVIEW OF MATRICES AND QUADRATIC FORMS

Recall also that, in accordance with the well-known theorem on multiplication of determinants, the matrix equation C = AB implies the determinant equation |C| = |A||B|. The matrix   1 0 ... 0   0 1 ... 0 E=   .............    0 0 ... 1 is called the unit matrix. It is easy to see that always AE = EA = A. A matrix B is called an inverse of A (B = A−1 ), if AB = E. It follows from AB = E that |A||B| = |E| = 1 and, consequently, |A| = 0. Conversely, it is easy to show that if |A| = 0, then for the matrix A, there exists one and only one inverse matrix B = bik n1 , with bik =

(−1)i+k A |A|



1 2 ... 1 2 ...

k − 1 k + 1 ... i − 1 i + 1 ...

n n

 (i, k = 1, 2, . . . , n).

We see from this that the equality AB = E implies the equality BA = E; i.e., from B = A−1 follows A = B −1 . We remark that the above definition of the product of two matrices is applicable also when the factors are rectangular matrices, provided the number of columns in the first factor is equal to the number of rows in the second one. The theorem of determinant multiplication is a special case of the following theorem of Binet-Cauchy1 concerning the multiplication of an elongated matrix by a foreshortened one. Let there be given two rectangular matrices    a11 . . . a1n     ...............  A=   ...............    am1 . . . amn

   b11 . . . b1m     ..............  B= ,  ..............    bn1 . . . bnm

and

(2)

where m ≤ n. The product AB of these matrices, is the square matrix C = cik m 1 , where n  cik = aij bjk (i, k = 1, 2, . . . , m). j=1

Then  1 2 ... m 1 2 ... m   1 2 ... = A α1 α2 . . . 

|C| = C

m αm



 B

α1 1

α2 2

... ...

αm m

 ,

where the sum in the right hand side is over all possible combinations (α1 . . . αm ) of m elements of the set 1, 2 . . . , n. If m < n and we make up the product of matrices (2) in the opposite order; i.e., we multiply the foreshortened matrix B by the elongated matrix A, and we 1 See,

for example, A.K. Sushkevich [55], Sec. 38, and also V.F. Kagan [24], Sec. 49.

1. MATRICES AND OPERATIONS ON MATRICES

11

obtain a square matrix D = dik n1 with elements dik =

m 

bin ank

(i, k = 1, 2, . . . , n),

n=1

the determinant of which is equal to zero:  1 2 ... |D| = D 1 2 ...

n n

 = 0.

Let now A = aik n1 and B = bik n1 be two square matrices of the same order and let C = AB. Then the matrix of the minor   i1 . . . ip C (p ≤ n) k1 . . . kp is the product of two rectangular matrices    ai 1 ai 2 . . . ai n  1 1   1  .....................     ai 1 ai 2 . . . ai n  p p p

and

   b1k1 . . . b1kp     b2k1 . . . b2kp   .  ...............    bnk1 . . . bnkp

Therefore, by the Binet–Cauchy theorem   i1 i2 . . . ip C k1 k2 . . . kp   i1 i2 . . . ip   α1 B = A α1 α2 . . . αp k1

α2 k2

... ...

αp kp

 ,

(3)

where the sum extends over all possible combinations (α1 α2 . . . αp ) of the n indices, 1, 2, . . . , n taken p at a time. This formula gives an expression for the minors of the product of two matrices in terms of the minors of the factors. Let us draw still another conclusion from this. Let rA , rB , and rC , denote respectively the ranks of the matrices A, B, and C. Then it follows from (3) that2 rC ≤ rA , rB . If the matrix B is non-singular, then A = CB −1 , and this means that rA ≤ rC , rB −1 ; hence r C = rA . Thus, when a square matrix is multiplied by a non-singular one, its rank does not change. 2 By r ≤ r , r C A B we denote in abbreviated form a system of two inequalities, rC ≤ rA and rC ≤ rB .

12

I. REVIEW OF MATRICES AND QUADRATIC FORMS

2. Sylvester’s identity In this section, we shall give a proof of a well-known proposition concerning the minors of the inverse matrix, and, on the basis of this proposition, we shall derive the Sylvester identity, which is hard to find in the popular texts. Modifying somewhat the ordinary definition, we shall call a matrix A˜ = ˜ aik n1 n the adjoint of a matrix A = aik 1 , if a ˜ik is the minor of the element aik ; i.e.,3  a ˜ik = A

i − 1 i + 1 ... k − 1 k + 1 ...

1 2 ... 1 2 ...

n n

 (i, k = 1, 2, . . . , n).

The following formula holds for the minors of an adjoint matrix:  A˜

i1 k1

i2 k2

... ...

ip kp



 = |A|p−1 A

j2 l2

j1 l1

... ...

jn−p ln−p

 ,

(4)

where the system of indices i1 < i2 < · · · < ip , j1 < j2 < · · · < jn−p , as well as the system of indices k1 < k2 < · · · < kp , l1 < l2 < · · · < ln−p coincides with the total system of indices 1, 2, . . . , n. We shall prove this formula for the minor  A˜

1 1

2 ... 2 ...

p p

 .

The reader can readily verify that the general case is obtained from this one by suitable rearrangement of the rows and columns in the matrix A. Let us introduce the algebraic complement Aik = (−1)i+k a ˜ik of the element aik(i, k = 1, 2, . . ., n). Multiplying the i-th row (i = 1, . . . , p) of the determinant 1 2 ... p by (−1)i , and the k-th column (k = 1, . . . , p) by (−1)k , we can A˜ 1 2 ... p represent this determinant in the following form:

 A˜

1 2 ... 1 2 ...

   A11 . . . A1p A1,p+1 . . . A1n     ..................................     Ap1 . . . App Ap,p+1 . . . Apn  p   = 0 ... 0 1 0 ... 0  p   ... 0 0 1 ... 0   0    ..................................    0 ... 0 0 0 ... 1

Let us multiply both sides of this equality by the determinant    a11 . . . a1n    |A| =  . . . . . . . . . . . . . .  ,  an1 . . . ann  3 In the usual definition, a ˜ik is the cofactor of the element aik ; i.e., the minor of the element aik multiplied by (−1)1+k .

2. SYLVESTER’S IDENTITY

13

and multiply the determinants in the right hand side horizontally; we then obtain   0 ... 0 0 ... 0   |A|   |A| ... 0 0 ... 0   0      ...................................................  1 2 . . . p   0 ... |A| 0 ... 0 . |A| =  0 A˜ 1 2 ... p    a1,p+1 a2,p+1 . . . ap,p+1 ap+1,p+1 . . . an,p+1     ...................................................    a1n a2n ... apn ap+1,n ... ann But the determinant in the right hand side is  p + 1 ... p |A| A p + 1 ... therefore, if |A| = 0, then  1 2 ... ˜ A 1 2 ...

p p



n n 

= |A|

p−1

A

 ;

p + 1 ... p + 1 ...

n n

 .

(5)

From continuity considerations it is clear that this equation also holds also when |A| = 0. Thus we have proved formula (4). Let us notice that when p = n, formula (4) assumes the form ˜ = |A|n−1 . |A|

(6)

From the formula for the minors of the adjoint matrix we obtain the well-known formula for the minors of the inverse matrix. If A = aik n1 and B = A−1 , then   l . . . l l 1 2 n−p p p   A   iν + k ν j1 j2 . . . jn−p i1 i2 . . . ip 1  ,  (7) = (−1) 1 B k1 k2 . . . kp 1 2 ... n A 1 2 ... n where i1 < i2 < · · · < ip together with j1 < j2 < · · · < jn−p , and k1 < k2 < · · · < kp together with l1 < l2 < · · · < ln−p comprise the total system of indices 1, 2, . . . , n. ˜ki /|A| (i, k = 1, 2, . . . , n). Indeed, B = bik n1 , where bik = Aki /|A| = (−1)i+k a Therefore   k . . . k k 1 2 p ˜ p p   A   iν + k ν i1 i2 . . . ip i1 i2 . . . ip 1 B = (−1) 1 , k1 k2 . . . kp |A|p and from this, using (4), we obtain (7). Theorem 1 (Sylvester). Let A = aik n1 be an arbitrary matrix and 1 ≤ p < n. We put   1 2 ... p i (i, k = p + 1, . . . , n) bik = A 1 2 ... p k and B = bik np+1 . Then    1 p + 1 ... n =A B 1 p + 1 ... n

2 2

... ...

p p

n−p−1

 A

1 1

2 2

... ...

n n

 .

(8)

14

I. REVIEW OF MATRICES AND QUADRATIC FORMS

Proof. By (4) we have   p + 1 ... r − 1 r + 1 ... n = A˜ p + 1 ... s − 1 s + 1 ... n   1 2 ... p r |A|n−p−2 = brs |A|n−p−2 . =A 1 2 ... p s On the other hand, the matrix crs np+1 , where  p + 1 ... r − 1 r + 1 ... crs = A˜ p + 1 ... s − 1 s+ 1 ...

n n

(9)

 .

is adjoint of the matrix ˜ aik np+1 . Consequently, by virtue of (6), we have  C

p + 1 ... p + 1 ...

But according to (9)  p + 1 ... C p + 1 ... and according to (4)  p + 1 ... ˜ A p + 1 ...

n n

n n

n n



 = A˜



 =B

p + 1 ... p + 1 ...

p + 1 ... p + 1 ...

n−p−1

 =A

1 1

2 ... 2 ...

n n

n n

n−p−1 .

(10)

 |A|(n−p−2)(n−p) ,

p p

n−p−1

2

|A|(n−p−1) .

Equating, the right hand sides of these equations, using (10), and dividing through by |A|(n−p−2)(n−p) (under the assumption that |A| = 0), we arrive at Sylvester’s identity (8). From continuity considerations, it follows that this identity also holds for |A| = 0. Henceforth we shall make use of the formula   i1 . . . iq B k1 . . . kq q−1   1 2 ... 1 2 ... p A =A 1 2 ... 1 2 ... p

p i1 p k1

... ...

iq kq

 ,

(11)

which obviously is the Sylvester identity for the matrix of the determinant   1 2 . . . p i1 . . . iq . A 1 2 . . . p k1 . . . kq 3. Eigenvalues and eigenvectors of a matrix 1. An ordered system of n complex numbers x1 , x2 , . . . , xn will be called a vector x, and the numbers xi themselves (i = 1, 2, . . . , n) the coordinates of this vector; we shall write x = (x1 , x2 , . . . , xn ). The vector (0, 0, . . . , 0) will be denoted by 0. Of any k vectors x = (x1 , x2 , . . . , xn ), y = (y1 , y2 , . . . , yn ), . . . , v = (v1 , v2 , . . . , vn )

3. EIGENVALUES AND EIGENVECTORS OF A MATRIX

15

and k numbers a, b, . . . , f , we can form a vector w = ax + by + · · · + f v, the coordinates of which are determined by the formula wi = axi + byi + · · · + f vi

(i = 1, 2, . . . , n).

The vectors x, y, . . . , v are called linearly dependent (independent) if there exists (there does not exist) such a system of numbers, not simultaneously equal to zero, a, b, . . . , f , that ax + by + · · · + f v = 0. From the theory of determinants it follows that for the k vectors x, y, . . . , v to be linearly independent, it is necessary and sufficient that the rank of the matrix    x 1 x2 . . . xn     y1 y2 . . . yn     ................    v1 v2 . . . vn formed of the coordinates of these vectors be equal to k. We shall say that the vector x transforms into a vector x by means of the matrix A = aik n1 , and we shall write x = Ax, if xi =

n 

aik xk

(i = 1, 2, . . . , n).

k=1

It is obvious that A(ax + by + · · · + f v) = aAx + bAy + · · · + f Av. Let, furthermore,

x = Ax and where A = aik n1 , B = bik n1 . Then xi =

n 

aij xj , xj =

j=1

n 

bjk xk

x = Bx,

(i, j = 1, 2, . . . , n);

k=1

hence xi =

n 

cik xk

(i = 1, 2, . . . , n),

k=1

where cik =

n 

aij bjk

(i, k = 1, 2, . . . , n);

j=1

i.e.,

(C = AB). x = Cx In other words, successive transformation of a vector first with the aid of the matrix B, and then with the aid of the matrix A, leads to the same result as the transformation of the vector with the aid of matrix C, equal to the product of the matrices A and B: A(Bx) = (AB)x. In particular (when AB = E), we obtain from this:

16

I. REVIEW OF MATRICES AND QUADRATIC FORMS

If |A| = 0, then x = Ax leads to x = A−1 x . A particular role in the investigation of matrices is played by those vectors, which are transformed by means of a given matrix into collinear vectors. If Ax = λx

(12)

and x = 0, then the vector x is called an eigenvector, and the number λ the corresponding eigenvalue or the characteristic number of the matrix A. The equation (12) can be written in coordinates in the form of the system of equations n  aik xk = λxi (i = 1, 2, . . . , n), k=1

or, with the aid of the Kronecker symbol δik (δik = 1 when i = k and δik = 0 when i = k), in the form n 

(aik − λδik )xk = 0

(i = 1, 2, . . . , n).

(13)

k=1

The latter system of equations has a non-trivial solution (x = 0) if and only if the determinant4 Δ(λ) = |aik − λδik |n1 = (−λ)n + S1 (−λ)n−1 + S2 (−λ)n−2 + · · · + Sn vanishes. Δ(k) is called the characteristic determinant, and the equation Δ(λ) = 0 the characteristic equation5 of the matrix A. Thus, a number λ is an eigenvalue of the matrix A if and only if it coincides with one of the roots of the characteristic equation. Since the characteristic equation is an algebraic equation of n-th degree, it has n (not necessarily distinct, generally speaking complex) roots. Consequently, every matrix has at least one eigenvector. 2. It is obvious that if a vector u is an eigenvector, then the vector cu (where c is a number different from zero) is also an eigenvector and corresponds to the same eigenvalue. Generally, if an eigenvalue λ0 corresponds to several eigenvectors u, v, . . . , w, then any linear combination of these vectors au + bv + · · · + f w, yields an eigenvector for the same eigenvalue λ0 , provided this combination differs from zero. The question arises: what is a maximal number d0 of linearly independent eigenvectors, corresponding to a given eigenvalue λ0 ? From the well-known theorem on the solutions of a system of linear homogeneous equations6 it follows that this number d0 equals the defect of the matrix of the system of equations (13) (when λ = λ0 ); i.e., to the defect of the matrix A − λ0 E = aik − λ0 δik n1 ; we recall that by the defect of a square matrix we understand the difference between its order and its rank.    aii aik   , etc.; i.e., Sk (k = 1, 2, . . . , n) is i 0, . . . , as > 0, as+1 < 0, . . . , ar < 0 Let some other representation of the form A(x, x) as a sum of independent squares be ρ  A(x, x) = bi Yi2 , (43) i=1

and the following inequalities hold b1 > 0, . . . , bσ > 0, bσ+1 < 0, . . . , bρ < 0. We shall prove the theorem by contradiction. Assume, for example, that σ < s; then ν = σ + (r − s) < r ≤ n, and consequently the system of ν < n equations with n unknowns x1 , x2 , . . . , xn Y1 = 0, Y2 = 0, . . . , Yσ = 0, Xs+1 = 0, . . . , Xr = 0

(44)

has non-trivial solutions. Let us consider, on the other hand, a system of equations X1 = 0, X2 = 0, . . . , Xr = 0.

(45)

Since it consists of r independent equations, it cannot be equivalent to a system of ν < r equations (44). Therefore there are such values x0i , that when x = x0i (i = 1, 2, . . . , n) the system (44) will be satisfied, and at the same time one of the Xj (j = 1, 2, . . . , s) will differ from zero. But then for x0 = (x01 , x02 , . . . , x0n ) we have, by (42), A(x0 , x0 ) > 0, and by (43) we have A(x0 , x0 ) ≤ 0. We have arrived at a contradiction. The Law of Inertia makes it possible to introduce the following terminology: The number of positive (or negative) numbers ai in the representation (42) will be called the number of positive (or negative) squares of the form A(x, x). Recalling Theorem 3, we arrive at the following conclusion: Rule. The number of positive (or negative) eigenvalues of a real symmetric matrix A is equal to the number of positive (or negative) squares of the form A(x, x). We remark furthermore that since the number of non-zero eigenvalues for a matrix of simple structure equals the rank of the matrix, in the representation (42) the number of squares r is nothing but the rank of the matrix A. 3. In determining the number of positive or negative squares of the form A(x, x), various devices can be used. All the principal devices can be obtained from Sylvester’s formula, which he discovered in 1851. 8 i.e.

the number of positive ai

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I. REVIEW OF MATRICES AND QUADRATIC FORMS

In this formula we shall use the abbreviated notation: Ai (x) =

n 

aik xk

(i = 1, 2, . . . , n).

k=1

Let A = aik n1 be a symmetric matrix, and   1 2 ... p  0 = Ap = A 1 2 ... p

Sylvester’s formula.

for some p (1 ≤ p ≤ n). Then the following identity holds A(x, x)

  ... a1p A1 (x)   a11  1  . . . . . . . . . . . . . . . . . . . . . . . . . . .  =−   ... app Ap (x)  Ap  ap1   A1 (x) . . . Ap (x) 0   n 1  1 2 ... p i x i xk . + A 1 2 ... p k Ap

(46)

i,k=1

Proof. Equation (46) is equivalent to the following:   ... a1p A1 (x)   a11  n    1 2 ...  ............................  A  = 1 2 ... ... app Ap (x)   ap1   i,k=1 A1 (x) . . . Ap (x) A(x, x)

p i p k

 x i xk .

To verify this equation, we rewrite its left hand side in the expanded form   n    ... a1,p a1k xk   a11   k=1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .n. . . . . . . . .      ap1 ... app apk xk   k=1   n n n      ai1 xi . . . aip xi aik xi xk   i=1

i=1

(47)

i,k=1

and represent the last row as a sum of n rows and the last column as a sum of n columns. Then, from the theorem on addition of determinants, the determinant (47) is represented as a sum of n2 determinants   a1p a1k xk   a1i . . .    ..........................   (i, k = 1, 2, . . . , n),  app apk xk   ap1 . . .   ai1 xi . . . aip xi aik xi xk   1 2 ... p i xi xk (i, k = 1, 2, . . . , n). which are equal to to A 1 2 ... p k We note that in the right hand side of Sylvester’s formula (46) the summation is extended only over the values  greater than p, for if i or k is  of i and k that are 1 2 ... p i = 0. less than or equal to p, then A 1 2 ... p k

6. REDUCTION OF A QUADRATIC FORM TO A SUM OF SQUARES

31

4. We now proceed to some specific methods of reducing a quadratic form to a sum of squares. The Lagrange method. This consists of a successive separation of one ofrtwo independent squares. Here it is necessary to distinguish two cases: 1) Let in a matrix A = aik n1 the element a11 = 0. Putting p = 1 in Sylvester’s formula (46), we find   n 1 1  1 i 2 x i xk . A(x, x) = [A1 (x)] + A (48) 1 k a11 a11 i,k=2

Thus, we have separated from the form A(x, x) one square in such a way that the remaining expression   n 1  1 i xi x k R(x, x) = A 1 k a11 i,k=2

contains only the variables x2 , x3 , . . . , xn . Therefore, if we succeed in representing R(x, x) as a sum of independent squares R(x, x) =

m 

ai Xi2

(ai = 0),

i=2

where Xi (i = 2, 3, . . . , m) depends only on x2 , x3 , . . . , xn , then, since A1 (x) contains x1 , the representation m  1 2 A(x, x) = [A1 (x)| + ai Xi2 a11 i=2

will be a representation of the form A(x, x) as a sum of independent squares. It is obvious that such a method of separating one square is applicable in general if any element agg on the principal diagonal (and not only a11 ) is different from zero. Indeed, if agg = 0, then   1 1   agg agk  2 A(x, x) = [Ag (x)] +  aig aik  xi xk . agg agg i,k=g

 0. We now put p = 2 in Sylvester’s formula 2) Let a11 = a22 = 0, but a12 = (46), and obtain    0   n a12 A1 (x)   1  1 1 2 i xi xk , A(x, x) = 2  a12 0 A2 (x)  − 2 A 1 2 k a12  a12 A1 (x) A2 (x) 0  i,k=3 or A(x, x) =

1 1 [A1 (x) + A2 (x)]2 − [A1 (x) − A2 (x)]2 + R, 2a12 2a12

where R=−

  n 1  1 2 i x i xk A 1 2 k a212 i,k=3

depends only on x3 , . . . , xn .

(49)

32

I. REVIEW OF MATRICES AND QUADRATIC FORMS

Since A1 (x) = a12 x2 + a13 x3 + · · · + a1n xn , A2 (x) = a21 x1 + a23 x3 + · · · + a2n xn , the functions A1 (x) and A2 (x) are independent, and consequently the first two squares in the representation (49) are also independent. Obviously this method of separating two squares is applicable in general if for certain g = h we have agg = ahh = 0, agh = 0, and then we have in place of (49) A(x, x) =

1 [Ag (x) + Ah (x)]2 2agh  1 1  g 2 − [Ag (x) − Ah (x)| − 2 A g 2agh agh

h i h k

 x i xk .

If the form A(x, x) does not vanish identically, then either for some g (1 ≤ g ≤ n) we have agg = 0, or a11 = a22 = · · · = ann = 0, and agh = 0 for some g and h (g = h). From this we conclude that by a successive combination of steps 1) and 2) we can reduce any form to a sum of squares using only rational operations. 5. The Jacobi method. Let us denote for brevity  a Δ0 = 1, Δ1 = a11 , Δ2 =  11 a21

 a12  , a22 

...,

   a11 . . . a1n    Δn =  . . . . . . . . . . . . . .  .  an1 . . . ann 

Jacobi noticed that if none of the successive principal minors Δk (k = 1, 2, . . . , n) vanishes, then n  Xk2 A(x, x) = , (50) Δk Δk−1 k=1

where    a11 . . . a1,k−1 A1 (x)     ........................  X1 = A1 (x), Xk =   (k = 2, 3, . . . , n).  ........................    ak1 . . . ak,k−1 Ak (x)

(51)

We shall prove a somewhat more general proposition: If a form A(x, x) has rank r and Δk = 0 (k = 1, 2, . . . , r), then A(x, x) =

r  k=1

Xk2 . Δk Δk−1

(52)

For this purpose we put p = r in Sylvester’s formula (46); we then obtain the identity known as the Kronecker identity:   ... a1r A1 (x)   a11  1  . . . . . . . . . . . . . . . . . . . . . . . . . .  A(x, x) = −  . ... arr Ar (x)  Δr  ar1   A1 (x) . . . Ar (x) 0

6. REDUCTION OF A QUADRATIC FORM TO A SUM OF SQUARES

33

Let us put   ... a1k A1 (x)   a11    ...........................  Pk (x, x) =   ... akk Ak (x)   ak1   A1 (x) . . . Ak (x) 0

(k = 1, 2, . . . , r).

For any matrix C = cij k+1 of order k + 1 we have the following identity: 1     1 2 ... k − 1 k + 1 1 2 ... k C C 1 2 ... k − 1 k + 1 1 2 ... k     1 2 ... k − 1 k + 1 1 2 ... k − 1 k C −C 1 2 ... k − 1 k 1 2 ... k − 1 k + 1     1 2 ... k k + 1 1 2 ... k − 1 . (53) C =C 1 2 ... k k + 1 1 2 ... k − 1  k−1 j Indeed, putting bjl = C (j, l = k, k + 1), we have in the k−  1 l k k+1 so that (53) is a special case left hand side of (53) the determinant B k k+1 of the Sylvester determinant identity, given in Sec. 2, equation (8). In particular, for a symmetric determinant Pk (x, x), we find from (53) that 

1 2 ... 1 2 ...

Pk−1 (x, x)Δk − Xk2 = Δk−1 Pk (x, x), or Xk2 Pk−1 (x, x) Pk (x, x) = − Δk Δk Δk−1 Δk−1 Hence, by the Kronecker’s identity we have −

(k = 1, 2, . . . , n; P0 = 0).

Xk2 Pr (x, x)  = . Δr Δk Δk−1 r

A(x, x) = −

k=1

This proves Jacobi’s formula. Inserting in (51) instead of Ai (x) (i = 1, 2, . . . , n) their expressions in terms of x1 , x2 , . . . , xn and using the theorem on the addition of determinants, we find that   n  a11 . . . a1,k−1 a1j     . . . . . . . . . . . . . . . . . . . . . .  xj = Δk xk + . . . (k = 1, 2, . . . , r). Xk =   j=k  ak1 . . . ak,k−1 akj  Since Δk = 0, the forms Xk (k = 1, 2, . . . , r) are linearly independent9 for if one goes backwards, then each new square contains a new variable. Thus, the Jacobi formula (52) gives the representation of the form A(x, x) as a sum of independent squares, and hence: 9 Incidentally, this follows directly from the fact that r is the rank of the form A(x, x), since were the Xk linearly dependent, then A(x, x) could be represented as a sum of squares using less than r terms; i.e., r could not be the rank of the form A(x, x).

34

I. REVIEW OF MATRICES AND QUADRATIC FORMS

Theorem 5. If a symmetric matrix A = aik n1 has the rank r and the successive principal minors Δ0 = 1, Δ1 , Δ2 , . . . , Δr (54) of this matrix are all different from zero, then the number of positive (or negative) squares of the form A(x, x) equals the number of sign changes of in the series (54). We remark that the Jacobi rule can be generalized to include the case when some terms (not the extreme ones) vanish in the series (54), but no two adjacent terms vanish simultaneously. In this case, if some Δk equals zero (1 < k < r), then Δk−1 and Δk+1 must have different signs. Indeed, applying the identity (53) to the , we find that matrix aij k+1 1  2    1 2 ... k − 1 k 1 2 ... k − 1 k + 1 − A Δk A 1 2 ... k − 1 k + 1 1 2 ... k − 1 k + 1 = Δk−1 Δk+1 . Consequently, if Δk = 0, Δk−1 = 0 and Δk+1 = 0, then Δk−1 Δk+1 < 0. In this case (when no adjacent terms in the series (54) vanish simultaneously) the number of sign changes in the series (54) is independent of the sign prescribed to the zero terms of the series (54). From continuity considerations it is easy to verify (we leave this to the reader) that Theorem 5 remains true also in this case. 7. Positive quadratic forms A real quadratic form A(x, x) =

n 

aik xi xk

i,k=1

is called non-negative if for every vector x, we have A(x, x) ≥ 0,

(55)

and positive, if for every vector x = (x1 , x2 , . . . , xn ) = 0, we have A(x, x) > 0.

(56)

From the representation of the form A(x, x) =

n 

λi ξi2 ,

i=1

where λi (i = 1, 2, . . . , n) are the eigenvalues of the matrix A (see Sec. 5), we derive the following: Theorem 6. In order for a form A(x, x) to be non-negative (or respectively positive), it is necessary and sufficient that all the eigenvalues λ1 , λ2 , . . . , λn of the matrix A be non-negative (respectively positive). Thus, a positive quadratic form A(x, x) can always be represented as a sum of n independent positive squares, and a non-negative form can be represented as a sum of r positive independent squares, where r is the rank of this form.

7. POSITIVE QUADRATIC FORMS

35

Theorem 7 (criterion of positivity of a quadratic form). In order for a quadratic form A(x, x) to be positive, it is necessary and sufficient that all successive principal minors of its matrix be positive; i.e.,    a11 . . . a1n       a11 a12   > 0, . . . ,  . . . . . . . . . . . . . .  > 0. a11 > 0,    a21 a22   an1 . . . ann  Proof. Let us put Δ0 = 1,

...

   a11 . . . a1k    Δk =  . . . . . . . . . . . . .   ak1 . . . akk 

(k = 1, 2, . . . , n).

If the quadratic form A(x, x) is positive, then all λi > 0 (i = 1, 2, . . . , n) and consequently the discriminant has the form Δn = λ1 λ2 . . . λn > 0. On the other hand, if the x1 , x2 , . . . , xp do not vanish simultaneously and xp+1 = · · · = xn = 0, we obtain from (56) that p 

(p = 1, 2, . . . , n − 1);

aik xi xk > 0

i,k=1

i.e., the “cut off” forms

p 

aik xi xk (p = 1, 2, . . . , n − 1) are also positive, and

i,k=1

therefore their discriminants satisfy Δp > 0 (p = 1, 2, . . . , n − 1). In the opposite direction, if Δ1 > 0, Δ2 > 0, . . . , Δn > 0,

(57)

we obtain from the Jacobi formula (50) A(x, x) =

n  i=1

Xi2 > 0, Δi Δi−1

provided that not all Xi = 0 (i = 1, 2, . . . , n). But by linear independence of the forms Xi (i = 1, 2, . . . , n) these forms vanish simultaneously only when xi = 0 (i = 1, 2, . . . , n). Corollary. If the successive principal minors (57) are positive in a symmetric matrix A = aik n1 then all its principal minors are positive. Indeed, it follows from (57) that the form A(x, x) is positive. On the other hand, by a suitable change of enumeration of the variables xi , i.e., the numbering of the rows and columns, we can include any principal minor in the chain of successive principal minors. Thus, all principal minors are positive. Theorem 8 (criterion of non-negativity of a quadratic form). In order for a quadratic form A(x, x) to be non-negative, it is necessary and sufficient that all the principal minors of the matrix A be non-negative.

36

I. REVIEW OF MATRICES AND QUADRATIC FORMS

Proof. Let us put Aε = A + εE = aik + εδik n1 Then Aε (x, x) =

n  i,k=1

aik xi xk + ε

(ε > 0). n 

x2i .

i=1

Consequently, if the form A(x, x) is non-negative, then the form Aε (x, x) is positive and this implies that its principal minors are positive. Since Aε → A when ε → 0, the principal minors of the matrix A are non-negative. Conversely, if all the principal minors of the matrix A are non-negative, then for the discriminant of the form Aε (x, x) for ε > 0 we have     a aik  n−2 ii n n n−1  |aik + εδik |1 = ε + aii ε + + ··· > 0  aki akk  ε and we conclude that all the principal minors of the matrix Aε are positive. Hence the form Aε (x, x) is positive, so the form A(x, x) = lim Aε (x, x) is non-negative. ε→0

Let us call the reader’s attention to the fact that from the non-negativity of the successive principal minors of a symmetric matrix A it does not follow that the form A(x, x) is non-negative, and consequently, the non-negativity of the other principal minors does not follow. n  Indeed, let us take for example an arbitrary form aik xi xk , which is not i,k=2

non-negative, and consider it as a quadratic form of n variables x1 , x2 , . . . , xn , putting a1i = ai1 = 0 (i = 1, 2, . . . , n). Then all successive minors   1 2 ... p (p = 1, 2, . . . , n) A 1 2 ... p will vanish (i.e., they will be non-negative), while the quadratic form

n 

aik xi xk

i,k=1

is not non-negative. 8. Hadamard’s inequality The simplest and at the same time the most accurate estimates of determinants were proposed first by Hadamard. Theorem 9 (Hadamard’s inequality). If a form A(x, x) is positive, then the following inequality holds for its discriminant    a11 . . . a1n     . . . . . . . . . . . . . .  ≤ a11 a22 . . . ann , (58)    an1 . . . ann  and equality holds if and only if aik = 0 for i = k (i, k = 1, 2, . . . , n).

8. HADAMARD’S INEQUALITY

Proof. To prove inequality (58) it is sufficient to show that     1 2 ... n − 1 1 2 ... n ann , ≤A A 1 2 ... n − 1 1 2 ... n

37

(59)

with equality taking place if and only if a1n = a2n = · · · = an−1,n = 0. Indeed, if the form A(x, x) is positive, then the cut-off forms p 

aik xi xk

(p = 1, 2, . . . , n)

i,k=1

are all positive. Applying inequality (59) to each of these, we obtain     1 2 ... n− 1 1 2 ... n ann ≤ ≤A A 1 2 ... n− 1 1 2 ... n   1 2 ... n − 2 an−1,n−1 ann ≤ · · · ≤ a11 a22 . . . ann . ≤A 1 2 ... n − 2

(60)

It is also obvious that equality can occur in (58) only when equality occurs in all relations (60), which leads to the conditions aik = 0 when i = k (i, k = 1, 2, . . . , n). To prove (59) let us consider all those δ, for which the form Aδ (x, x) = A(x, x) − δx2n is positive. According to the criterion for the positivity of a quadratic form, Aδ (x, x) is positive if the successive principal minors of the matrix Aδ are positive. But all these minors, except the last one, coincide with the corresponding minors of A, which by assumption (using the same criterion) are positive. Thus, the δ considered should satisfy the single condition    a11 . . . a1n    . . . . . . . . . . . . . . . . . .  = Δn − δΔn−1 > 0.    an1 . . . ann − δ  Δn is the least upper bound of the values of δ being Δn−1 considered. On the other hand, if the form Aδ (x, x) is positive, then all principal minors of its matrix are positive and, in particular, ann − δ > 0. Thus, the quantity ann is an upper bound (but generally not the least upper bound) for the numbers δ, and so Δn ≤ ann or Δn ≤ Δn−1 ann , (61) Δn−1 which gives the inequality (59). Let us now determine, when equality is possible in (59). If we have equality in (59), then ann is the least upper bound of those δ for which the form Aδ (x, x) is positive. Consequently, the form Consequently, the quantity

B(x, x) = A(x, x) − ann x2n , which is obtained from the form Aδ (x, x) with δ = ann , is non-negative. But then the principal minors of the matrix B      bkk bkn   akk akn   = −a2  = kn (k = 1, 2, . . . , n − 1)  bnk bnn   ank 0 

38

I. REVIEW OF MATRICES AND QUADRATIC FORMS

are non-negative, and consequently for k = 1, 2, . . . , n − 1.

akn = 0

Thus, the theorem is completely proved. Inequality (59), from which the Hadamard inequality (58) was obtained, is a special case of the more general inequality, to be established in the following theorem: n 

Theorem 10 (generalized Hadamard’s inequality).

If a form A(x, x) =

aik xi xk is positive, then for every p < n its discriminant satisfies:

i,k=1

 A

1 1

2 2

... ...

n n



 ≤A

1 1

2 2

... ...

p p



 A

p+1 p+1

... ...

n n

 ,

(62)

with equality if and only if aik = aki = 0

for

i = 1, 2, . . . , p;

k = p + 1, . . . , n.

(63)

Proof. The proof of this theorem does not depend on the proof of the preceding theorem; thereby we obtain a new proof of Theorem 9. The validity of the theorem for n = 2 can be seen directly, for in this case p = 1 and   1 2 = a11 a22 − a212 ≤ a11 a22 , A 1 2 with equality if and only if a12 = 0. Let us prove the theorem by induction with respect to n. If n > 2, then one of the numbers p, or n − p is greater than 1. Without loss of generality, we can assume that p > 1. We put   1 2 ... p − 1 i bik = A (i, k = p, p + 1, . . . , n). 1 2 ... p − 1 k Since by Sylvester’s identity (see Sec. 2, Theorem 1), for p ≤ q ≤ n we have  q−p      1 2 ... p − 1 p p + 1 ... q 1 2 ... q = B > 0, A 1 2 ... p − 1 p p + 1 ... q 1 2 ... q the form B(x, x) =

n 

bik xi xk

i,k=p

is positive. Consequently, by the  p p + 1 ... B p p + 1 ...  1 2 ... =A 1 2 ...  1 2 ... ×A 1 2 ...

induction assumption    n p + 1 ... n ≤ bpp B = n p + 1 ... n  n−p−1   1 2 ... p − 1 p 1 2 ... p − 1 p  p − 1 p + 1 ... n . p − 1 p + 1 ... n

(64)

8. HADAMARD’S INEQUALITY

39

On the other hand, if we put xp = 0 in the form A(x, x), we obtain again a positive form of n − 1 variables x1 , x2 , . . . , xp−1 , xp+1 , . . . , xn with discriminant   1 2 ... p − 1 p + 1 ... n . A 1 2 ... p − 1 p + 1 ... n Applying the generalized Hadamard inequality to this discriminant (which is possible by the induction assumption), we find that   1 2 ... p − 1 p + 1 ... n A 1 2 ... p − 1 p + 1 ... n     p + 1 ... n 1 2 ... p − 1 . (65) A ≤A p + 1 ... n 1 2 ... p − 1 Consequently  p p+1 B p p+1

 1 2 ≤ A 1 2



... ...

n n

... ...

p−1 p−1

 n−p  1 2 ... A 1 2 ...

and since by the Sylvester identity we have    1 2 ... p p + 1 ... n = A B 1 2 ... p p + 1 ... n we arrive to the required inequality    1 2 ... 1 2 ... n ≤A A 1 2 ... 1 2 ... n

p p

p−1 p−1

p p



 A

p + 1 ... p + 1 ...

 n−p  1 2 A 1 2



 A

p + 1 ... p + 1 ...

n n

... ...

n n

 ,

n n

 ,

 .

Here equality is possible only if equality holds in (64) and (65). But if in (65) we have equality, then by the induction assumption we have aik = aki = 0 for

i = 1, 2, . . . , p − 1 and k = p + 1, . . . , n.

(66)

On the other hand, if equality holds in (64), then by the induction assumption we have   1 2 ... p − 1 k = 0 for k = p + 1, p + 2, . . . , n. bpk = bkp = A 1 2 ... p − 1 p But by (66), expanding the determinant bkp with respect to the last row, we obtain     1 2 ... p − 1 1 2 ... p − 1 k (k = p + 1, . . . , n), = akp A A 1 2 ... p − 1 1 2 ... p − 1 p and therefore akp = apk = 0

for k = p + 1, . . . , n.

(67)

Conditions (66) and (67) together imply (63). This proves the theorem. Usually, it is not (58) that is called Hadamard’s inequality but the inequality from the following proposition:

40

I. REVIEW OF MATRICES AND QUADRATIC FORMS

Theorem 9 . For each real non-singular matrix C = cik n1 the following inequality holds:    c11 . . . c1n 2 n n n      2 2  ..............  ≤ c · c · · · c2nk , (68) 1k 2k    cn1 . . . cnn  k=1 k=1 k=1 and equality holds if and only if the rows of the matrix C are orthogonal to each other; i.e., if n 

cik cjk = 0

for

i = j

(i, j = 1, 2, . . . , n)10

(69)

k=1

Proof. We shall show that inequality (58) implies (68), and vice versa. For this purpose, starting with a matrix C = cik n1 , we put aij =

n 

cik cjk

(i, j = 1, 2, . . . , n).

(70)

k=1

Since such a definition of the numbers aij makes the form n 

aij xi xj =

i,j=1

n 

(c1k x1 + c2k x2 + · · · + cnk xn )2

(71)

k=1

positive, inequality (58) holds for the matrix A = aij n1 . By the theorem on multiplication of determinants  2    1 2 ... n 1 2 ... n = C A , (72) 1 2 ... n 1 2 ... n and consequently, (58) coincides with (68). In addition, by (70), conditions aij = 0 for i = j (i, j = 1, 2, . . . , n) coincide with (69). In the opposite direction, since any positive form A(x, x) can be represented as a sum of n linearly independent squares with positive coefficients and consequently, in the form (71) with |cik |n1 = 0, we obtain (58) from (68). Inequality (68) admits a simple geometric interpretation. Let us consider n linearly independent n-dimensional vectors ck = (ck1 , ck2 , . . . , ckn ) (k = 1, 2, . . . , n). Then the determinant |cik |n1 turns out to be equal to the signed volume of an oriented hyperparallelepiped, constructed on these vectors, and

 n  c2ik (i = 1, 2, . . . , n) k=1 10 Inequality (68) remains true also for non-singular matrices C = c n with complex ik 1 elements, provided the squares in both sides are replaced by the squares of the moduli; in this case, condition (69) must be replaced by n  k=1

cik cjk = 0

for

i = j (i, j = 1, 2, . . . , n).

8. HADAMARD’S INEQUALITY

41

gives the length of the vector ci (i = 1, 2, . . . , n). Thus, inequality (68) acquires the following simple geometric meaning: The volume of a hyperparallelepiped constructed on n vectors does not exceed the product of the lengths of these vectors. Equality holds if and only if all vectors are pairwise orthogonal. We shall leave it to the reader to state the geometric interpretation of the more general inequality (62). Let us note that (70) in matrix form can be written A = CC  .

(73)

Starting with this equality, we can obtain a new proof of inequality (62) and consequently a third proof of the Hadamard inequality. Let us show how this is done. First we notice that from the Binet–Cauchy theorem on the multiplication of rectangular matrices (see Sec. 1) and (73) follows that  

  2  1 2 ... p 1 2 ... p = A C , (74) 1 2 ... p i1 i2 . . . ip 1≤i1 0).

   ρk d Δ 1 = > 0, dλ Δ1 (μk − λ)2

1 Δ increases in each interval that does not contain the numbers μk (k = Δ1 1, 2, . . . , m). On the other hand, when λ changes from μk to μk+1 (k = 1, . . . , m−1), 1 Δ changes from −∞ to +∞. Consequently, within each such interval, the ratio Δ1  1 (λ) has one and only one root μ the polynomial Δ ˜k (k = 1, . . . , m − 1). Since, the ratio

58

I. REVIEW OF MATRICES AND QUADRATIC FORMS

1 Δ changes from −∞ to 1 when λ changes from μm to +∞ Δ1  1 (λ) has still another root μ then the polynomial Δ ˜m in the interval (μm , ∞). Thus furthermore, the ratio

˜ 1 < μ2 < μ ˜ 2 < · · · < μm < μ ˜m . μ1 < μ

(127)

Since the sequences λ1 ≤ λ2 ≤ · · · ≤ λ n ;

1 ≤ λ 2 ≤ · · · ≤ λ n , λ

when the numbers common to them are removed, become the sequences μ 1 < μ2 < · · · < μ m ,

μ ˜1 < μ ˜2 < · · · < μ ˜m ,

i follow from (127). the desired inequalities between the numbers λi and λ Since at least one of the numbers αi (i = 1, 2, . . . , n) does not vanish, then h > λh . The same m ≥ 1, and we conclude that there is a number of h such that λ can be obtained from the general minimax propositions. Quite similarly (and again by two methods), we can prove the following: ˜ x) is obtained from a form C(x, x) by the addition Theorem 18. If a form C(x, i (i = 1, 2, . . . , n) of the equations of r squares, then the successive roots λi and λ ˜ |A − λC| = 0 and |A − λC| = 0 satisfy the inequalities  h ≤ λh λ

(h = 1, 2, . . . , n),

h λn−r ≤ λ

(h = r + 1, . . . , n).

As in the case of the preceding theorem, one can state in addition that there  h < λh . is always such a value of h for which λ The next theorem can be considered a generalization of Theorem 15. Theorem 19. Let λ1 ≤ λ2 ≤ · · · ≤ λn be the eigenvalues of a pencil A(x, x) − λC(x, x), and λ∗1 ≤ λ∗2 ≤ · · · ≤ λ∗n−p the eigenvalues of the same pencil obtained upon superposition of p independent linear restrictions. Then the numbers λ∗1 , λ∗2 , . . . , λ∗n−p are not smaller than the corresponding numbers in the series λ1 , λ2 , . . . , λn−p and not greater than the corresponding numbers of the series λp+1 , . . . , λn : (k = 1, 2, . . . , n − p). λh ≤ λ∗h ≤ λp+h Proof. Let there be given p independent restrictions Λi (x) = 0 (i = 1, 2, . . . , p), and let xi =

n−p 

αik zk

(i = 1, 2, . . . , n)

(128)

k=1

be their parametric formulation. We denote by A∗ (z, z) and C ∗ (z, z) the forms obtained from A(x, x) and C(x, x) upon substitution of the expressions derived from (128) for the values of xi (i = 1, 2, . . . , n). We then have from Theorem 15   A∗ ∗ ∗ ∗ (h = 1, 2, . . . , n − p), λh = max μ L1 , . . . , Lh−1 ; ∗ C but it is obvious that to each h − 1 restrictions of the form L∗i (z) = 0 (i = 1, 2, . . . , h − 1)

10. MINIMAX PROPERTIES OF EIGENVALUES OF A PENCIL OF FORMS

59

correspond equivalent restrictions of the form Li (x) = 0 (i = 1, 2, . . . , h − 1), and vice versa. Furthermore, we have     A∗ A . μ L∗1 , . . . , L∗h−1 ; ∗ = μ L1 , . . . , Lh−1 , Λ1 , . . . , Λp ; C C Thus,

  A λ∗h = max μ L1 , . . . , Lh−1 ; Λ1 , . . . , Λp ; C

(129)

(We vary Li (i = 1, . . . , h − 1)). But by Theorem 15   A ≤ λh+p , μ L1 , . . . , Lh−1 ; Λ1 , . . . , Λp ; C consequently λ∗h ≤ λh+p

(h = 1, . . . , n − p).

On the other hand, since the minimum decreases as the number of restrictions decreases,     A A μ L1 , . . . , Lh ; Λ1 , . . . , Λp ; ≥ μ L 1 , . . . , Lh ; , C C and consequently, by (129) and Theorem 15,   A ∗ λh ≥ max μ L1 , . . . , Lh−1 ; = λh . C 4. We use the proved theorem to clarify the mutual position of the roots of the polynomials Δk (λ) = |aij − λcij |k1 (k = 1, 2, . . . , n). Since together with the form C(x, x) all the cut-off forms k 

cij xi xj

(k = 1, 2, . . . , n)

i,j=1

are also positive, the secular determinants Δk (λ) (k = 1, 2, . . . , n) have only real (k) (k) (k) roots. We denote them by λ1 ≤ λ2 ≤ · · · ≤ λk (k = 1, 2, . . . , n). The following proposition holds: Theorem 20. Roots of two successive polynomials Δk (λ) (k = 1, 2, . . . , n) alternate, i.e., (k)

(k−1)

λ1 ≤ λ1

(k)

(k−1)

≤ λ2 ≤ λ2

(k−1)

(k)

≤ · · · ≤ λk−1 ≤ λk

(k = 2, 3, . . . , n).

(130)

60

I. REVIEW OF MATRICES AND QUADRATIC FORMS (k)

Proof. The numbers λi

(i = 1, . . . , k) are the eigenvalues of the pencil k 

(aij − λcij )xi xj ,

i,j=1 (k−1)

and the numbers λi (i = 1, 2, . . . , k − 1) are the eigenvalues of the same pencil, but with one restriction xk = 0 imposed. Therefore the relation (130) is a simple consequence of the preceding theorem. The polynomial Δk (λ) can be represented in the following form: (k)

(k)

Δk (λ) = |cij |k1 (λ1 − λ) . . . (λk − λ) (k = 1, 2, . . . , n).

(131)

If the polynomials Δk (λ) and Δk−1 (λ) have no common roots (and for this it is necessary, but not sufficient, that Δk (λ) have no multiple roots), and only in this (k) (k−1) case, the numbers λi and λi strictly alternate, i.e., (k)

(k−1)

λ1 < λ1

(k)

(k−1)

< λ2 < λ2

(k−1)

(k)

< · · · < λk−1 < λk .

In this case it follows from (131) and from the inequality |cij |k1 > 0 (k = 1, 2, . . . , n) that when one of the intermediate polynomials Δn (λ), Δn−1 (λ), . . . , Δ1 (λ), Δ0 (λ) = 1

(132)

vanishes, the neighboring polynomials have opposite signs14 . Furthermore, as lim (−1)k Δk (λ) = ∞, in this case (and only in this case) the series (132) is a λ→∞

Sturm series (concerning Sturm series, see Sec. 1 of Chapter II). In the general case, the series (132) is not a Sturm series.15 11. Reduction of a matrix to a triangular form In this and the following sections we again return to consideration of arbitrary matrices (generally speaking, non-symmetric) with complex elements. A matrix A is called similar to a matrix B (A ∼ B), if there exists a nonsingular matrix P (|P | = 0) such that A = P BP −1 .

(133)

If A ∼ B, then B ∼ A, for it follows from (133) that B = P1 AP1−1 ,

where P1 = P −1 .

If A ∼ B, and B ∼ C, then A ∼ C. Indeed, from the equations A = P BP −1 , follows that

14 The

A = RCR−1 ,

B = QCQ−1 where

R = P Q.

same also follows from the generalized Jacobi rule (see the end of Sec. 6). in his well-known book “Analytical Dynamics” Russian translation, 1937, p. 207, having imprecisely formulated the generalized Jacobi rule, concludes that the series (132) is a Sturm series if the roots of the polynomial Δn (λ) are all simple. But this is not true, even under the assumption that all roots of all polynomials Δk (λ) are simple. 15 Whittaker,

11. REDUCTION OF A MATRIX TO A TRIANGULAR FORM

61

If A ∼ B, then A − λE = P (B − λE)P −1 , and consequently |A − λE| = |B − λE|, i.e., similar matrices have identical characteristic polynomials. A matrix T = tik n1 will be called triangular if tik = 0 when i > k, i.e.,    t11 t12 . . . t1n     0 t22 . . . t2n  T = .  ..................    0 0 . . . tnn In Sec. 3 we have seen that any matrix of simple structure is similar to some diagonal matrix λi δik n1 . Theorem 21. Every matrix is similar to a triangular matrix. Proof. We shall do the proof by induction. When n = 1 the statement is trivial. Let us assume that our statement is true for all matrices of order (n − 1). We shall prove that it is true for every matrix A = aik n1 . Let u = (u1 , . . . , un ) be an eigenvector of the matrix A: n 

aik uk = λui

(i = 1, 2, . . . , n).

(134)

k=1

We consider a non-singular matrix P = pik n1 , in which the first column consists of the coordinates of the vector u, i.e., pk1 = uk

(k = 1, 2, . . . , n).

(135)

Let B = P −1 AP = bik n1 ,

P −1 = pik n1 . (−1)

Then, by (134) and (135) we have bi1 =

n 

(−1)

pij

ajk pk1 =

n  j=1

j,k=1

p−1 ij

 n

 ajk uk

k=1

=λ

n 

(−1)

pij

pj1 = λδi1 .

j=1

Consequently, A ∼ B, where    λ b12 b13 . . . b1n     0 b22 b23 . . . b2n  B= .  ......................    0 bn2 bn3 . . . bnn Consider the matrix B1 = bik n2 of order (n − 1). By the induction assumption, there exists a non-singular matrix Q1 such that Q−1 1 B1 Q1 = T1 ,

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I. REVIEW OF MATRICES AND QUADRATIC FORMS

where T1 is a triangular matrix of order (n − 1). Let us put    1 1 0 0 ... 0     0 | 0 |     · | · | −1   ; then Q =  Q=   · | Q−1   · | Q1  1  · | · |     0 | 0 | and as can be readily seen  1 0 ...  0 |  · | Q−1 BQ =   · | Q−1  1 · |  0 |

    0 λ ∗ ... ∗ 1      0 |  0 |      · |  · | · ·  · |  · | B 1      · |  · |      0 |  0 |   λ ∗ ... ∗   0 |    · |    ; · | T  1   · |    0 | 

...

0

Q1

 0          

...

 0     =     

Here the asterisks denote the elements of no interest to us. Since T1 is a triangular matrix, it follows from the last equation that the matrix T = Q−1 BQ is also triangular. But A ∼ B, and B ∼ T ; consequently, A ∼ T , and our statement is proved. If A is similar to a triangular matrix T , then |A − λE| = |T − λE| = (t11 − λ) . . . (tnn − λ). Consequently, the numbers t11 , t22 , . . . , tnn always give a complete system of roots of the characteristic polynomial of the matrix A and therefore are determined uniquely by the matrix A. Thus we have proved that each matrix A admits of a representation of the form    λ1 ∗ ∗ . . . ∗     0 λ2 ∗ . . . ∗  −1 A=P (136)  P ,  ....................    0 0 0 . . . λn where P is a non-singular matrix and λ1 , . . . , λn is the complete system of eigenvalues of the matrix A. The representation (136) is not unique, and can be further specialized either by imposing restrictions on the elements denoted by the asterisks (for example, a triangular matrix can be reduced in the so-called Jordan normal form16 ), or by imposing restrictions on the matrix P (for example, by the Schur theorem17 one can choose P to be a so-called unitary matrix). We shall not dwell on all this, since we have no further use for any specialization of the representation (136). 16 See, 17 P.A.

for example, I.M. Gel’fand [17], Chapter III and A.I. Mal’tsev [37] Chapter IV. Shirokov [51] pp. 428-429.

12. POLYNOMIALS OF MATRICES

63

12. Polynomials of matrices We shall use in what follows the connection existing between the eigenvalues and the eigenvectors of a given matrix and of its power. Theorem 22. Let the numbers λ1 , . . . , λn form a complete system of eigenm  values of a matrix A = aik n1 and let f (x) = ck xk (cm = 0) be an arbitrary k=0

polynomial; then the numbers f (λ1 ), f (λ2 ), . . . , f (λn ) form a complete system of m  eigenvalues of the matrix f (A) = ck Ak (A0 = E). If, furthermore, the matrix k=0

A has simple structure, then the matrix f (A) is also simple; a fundamental matrix for A is also a fundamental matrix for f (A).18 Proof. As it can be readily verified, (136) implies that   ∗ ... ∗   f (λ1 )   f (λ2 ) . . . ∗  −1  0 f (A) = P  P ,  ..........................    0 0 . . . f (λn )

(137)

and from this, as already clarified in Section 11, it follows that the numbers f (λ1 ), f (λ2 ), . . . , f (λn ) form a complete system of eigenvalues of the matrix f (A). The second part of the theorem follows from the fact that, in the case of a matrix of simple structure, all elements marked with asterisks in (136), and in (137), will vanish if the matrix P is chosen to be a fundamental matrix for A. Let us give another proof of the first part of Theorem 22. This proof belongs to Frobenius and makes no use of the representation (136). Let us factor the given polynomial f (x) (in which, without loss of generality, we can assume the highest-order coefficient to be equal to one), together with the characteristic polynomial of the matrix, into linear factors f (x) = (x − α1 )(x − α2 ) . . . (x − am ), ϕ(x) = |A − xE| = (λ1 − x)(λ2 − x) . . . (λn − x).

(138) (139)

Inserting into (138) A in place of x we obtain f (A) = (A − α1 E)(A − α2 E) . . . (A − αm E), from which, taking (139) into account, we get |f (A)| = |A − α1 E||A − α2 E| . . . |A − αm E| =

 (λi − αk ). i,k

The latter equality shows that the determinant |f (A)| is the resultant of the polynomials f (x) and ϕ(x). This resultant, by virtue of (138), can be written |f (A)| = f (λ1 )f (λ2 ) . . . f (λn ).

(140)

Equation (140) is proved for any polynomial f (x). Replacing f (x) in this equation by f (x) − λ, we obtain an expression for the characteristic polynomial of the matrix f (A): |f (A) − λE| = [f (λ1 ) − λ][f (λ2 ) − λ] . . . [f (λn ) − λ], 18 We

notice that the converse is not always true.

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I. REVIEW OF MATRICES AND QUADRATIC FORMS

from which it follows directly that the numbers f (λ1 ), . . . , f (λn ) form a complete system of eigenvalues of the matrix f (A). 13. Associated matrices and the Kronecker theorem Let A = aik n1 be a matrix of order n. We introduce the concept of the p-th associated matrix Ap of the matrix A. For this purpose we consider all combinations (subsets of cardinality p) (i1 , i2 , . . . , ip ) of n indices 1, 2, . . . , n, and write each combination in the increasing order: i1 < i2 < · · · < ip . We order all combinations in such a way, so that a combination (i1 , i2 , . . . , ip ) precedes a combination (k1 , k2 , . . . , kp ) if and only if the first nonvanishing difference in the sequence k1 − i1 , k2 − i2 , . . . , kp − ip is positive. Thus, each combination (i1 , i2 , . . . , ip ) will occupy a definite place and consequently it will have  a definite number s, which can run through the values n . Let, for example, n = 5, p = 3; we then obtain the 1, 2, . . . , N , where N = p following series of combinations: (123), (124), (125), (134), (135), (145), (234), (235), (245), (345). Thus, the combination (145) has number 6, and the combination (245) has number 9. For the minors of p-th order of the matrix A we introduce the notation   i1 i2 . . . ip ast = A (i1 < i2 < · · · < ip ; k1 < k2 < · · · < kp ), k1 k2 . . . kp where s and t are respectively the numbers of the combinations (i1 , . . . , ip ) and (k1 , . . . , kp ). The matrix Ap = ast N 1 we be called the p-th associated matrix of A. Let C = AB, then, as we know from Sec. 1        i1 . . . ip i1 . . . ip α1 . . . αp C = A B . (141) k1 . . . kp α1 . . . αp k1 . . . kp We introduce the matrices Ap , Bp , Cp associated with the matrices A, B, and C. Then the relation (141) can be written as cst =

N 

asr brt ,

r=1

where s, t, and r are respectively the numbers of combinations (i1 , . . . , ip ),

(k1 , . . . , kp )

and

(α1 , . . . , αp ).

Consequently Cp = Ap Bp . Thus we have the following: 1◦ . The matrix associated with the product equals the product of the matrices associated with the factors.

13. ASSOCIATED MATRICES AND THE KRONECKER THEOREM

65

We shall apply this proposition to two mutually inverse matrices A and B (AB = E). We then obtain Ap Bp = Ep . But it is easily seen that Ep , the p-th associated matrix of E, is also a unit matrix: Cp = δst N 1 , and thus: 2◦ . The matrix associated with the inverse matrix, equals the inverse of the associated matrix. We now prove the following theorem: Theorem 23 (Kronecker). Let λ1 , λ2 , . . . , λn be the complete system of eigenvalues of the matrix A. Then the complete system of eigenvalues of the associated matrix Ap consists of all possible products of the numbers λ1 , . . . , λn , taken p at a time. Proof. According to Sec. 11, the matrix A admits a representation A = P T P −1 ,

(142)

where

   λ1 ∗ . . . ∗     0 λ2 . . . ∗  T = .  ................    0 0 . . . λn Using propositions 1◦ and 2◦ , we obtain from (142) A = Pp Tp P−1 p .

We can readily see that Tp = tst N 1 is a triangular matrix. Indeed, let   i1 . . . ip tst = T k1 . . . kp and s > t. The latter condition means that for some r < p i1 = k1 , . . . , ir = kr , But then

 T

i1 k1

i2 k2

... ...

ir+1 > kr+1 .

  ∗ ... ∗ ∗ ... ∗  λ i1   ∗ ∗ ... ∗  0 λ i2 . . .    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ip   0 . . . λir ∗ . . . ∗  = 0. = 0 kp   0 ... 0 0 ... ∗  0    ..............................    0 0 ... 0 0 ... 0

For the elements tss on the main diagonal, we have   i1 . . . ip = λ i1 λ i2 . . . λ ip . tss = T i1 . . . ip Since the elements of the main diagonal of the triangular matrix Tp give a complete system of eigenvalues of the matrix Ap , the Kronecker theorem is proved. For matrices of simple structure, the Kronecker theorem can be substantially supplemented. Any matrix A of simple structure admits a representation A = U λi δik n1 U −1 ,

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I. REVIEW OF MATRICES AND QUADRATIC FORMS

where U is a fundamental matrix for A. Passing in this relation to associated matrices, we obtain −1 Ap = Up Λs δst N 1 Up , where Λs = λi1 , . . . λip , and s is the number of the combination (i1 , i2 , . . . , ip ). Thus 3◦ . The matrix Up , associated with a fundamental matrix, is a fundamental matrix for the associated matrix Ap . From this we have the corollary: 4◦ . For a fixed combination (k1 , k2 , . . . , kp ) and a variable combination (α1 , α2 , . . . , αp ),  α1 α2 . . . αp k1 k2 . . . kp give all the coordinates of the eigenvector of the matrix Ap corresponding to the eigenvalue λk1 . . . λkp . 

the minors

U

http://dx.doi.org/10.1090/chel/345/03

CHAPTER II

Oscillatory Matrices We shall henceforth consider only matrices with real elements. 1. Jacobi matrices Before we study oscillatory matrices of general type, we shall consider normal Jacobi matrices. For these matrices one can, independently of the general theory, derive by elementary means all the basic properties that characterize oscillatory matrices. In addition, a study of Jacobi matrices is also of independent interest because these matrices play an important role in the investigation of small oscillations of various mechanical systems (for example torsional oscillations of a system of disks fastened to a shaft), and also in the theory of orthogonal polynomials, the problem of moments, etc. 1. A matrix J = aik n1 is called a Jacobi matrix if aik = 0 when |i − k| > 1. Introducing the notation ai = aii (i = 1, 2, . . . , n), bi = −ai,i+1 , ci = −ai+1,i (i = 1, 2, . . . , n − 1), we can write the matrix J in the form   0 0 ... 0   a1 −b1   0 ... 0   −c1 a2 −b2   −c2 a3 −b3 ... 0   0 J = .  .....................................     . . . . . . . . . . . . . . . . . . . . . . . . . . . . −bn−1    0 0 −cn−1 an

(1)

Let us put D0 (λ) ≡ 1,

Dk (λ) = |aij − λδij |k1

(k = 1, 2, . . . , n).

Thus, Dn (λ) is the characteristic polynomial of the matrix J. It is easy to see that the following recurrence formula holds for the polynomials Dk (λ) Dk (λ) = (ak − λ)Dk−1 (λ) − bk−1 ck−1 Dk−2 (λ) (k = 2, 3, . . . ).

(2)

Since D0 ≡ 1, and D1 (λ) = a1 − λ, it is possible to use this formula to calculate successively all the polynomials Dk (λ) (k = 2, 3, . . . , n). From this we conclude that in the expression Dk (λ) (k = 0, 1, . . . , n) the numbers bk and ck (k = 1, 2, . . . , n − 1) enter only in the form of products bk ck (k = 1, 2, . . . , n − 1). We shall confine our arguments only to the case when bk ck > 0 (k = 1, 2, . . . , n − 1). 67

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II. OSCILLATORY MATRICES

In this case, by the recurrence formula (2), the sequence of polynomials Dm , Dm−1 , . . . , D0

(m ≤ n)

has the first two properties of a Sturm sequence: 1◦ . D0 (λ) has constant sign (D0 ≡ 1). 2◦ . When Dk (λ) vanishes (1 < k < m) the polynomials Dk−1 (λ) and Dk+1 (λ) differ from zero and have opposite signs. Sturm has shown that when conditions 1◦ and 2◦ are satisfied, the increase in the number of sign changes in the sequence Dm , Dm−1 , . . . , D0 ,

(3)

as λ runs from α to β (α < β, Dm (α) = 0, Dm (β) = 0), is equal to the difference between the number of roots of the polynomial Dm (λ) on (α, β) at which the product Dm (λ)Dm−1 (λ) changes sign from + to −1 , and the number of roots of the polynomial Dm (λ) on the same interval at which the product Dm (λ)Dm−1 (λ) changes the sign from − to +. Let us give a simple argument that establishes Sturm’s rule2 . As λ changes from α to β, the number of sign changes in the sequence (3) can change only when λ passes through roots of the polynomials Dk (λ) (k = 1, . . . , m). However, if one of the intermediate functions Dk (λ) (1 < k < m) vanishes at λ0 (α < λ0 < β), then the two adjacent polynomials assume values with opposite signs at λ = λ0 ; consequently, when λ is sufficiently close to λ0 (greater and smaller than λ0 ) the sequence Dk+1 (λ), Dk (λ), Dk−1 (λ) will contain exactly one sign change. Thus, the number of sign changes in the sequence (3) can be affected only by passing through a root of Dm (λ), and obviously, if in such a transition the product Dm (λ)Dm−1 (λ) reverses sign from + to −, then one sign change is added in the sequence (3) (owing to the first two terms of the sequence Dm and Dm−1 ); if the product reverses sign from − to +, then one sign change is lost in the sequence (3); finally, if this product does not reverse sign, then the number of sign changes in (3) remains the same. Let us apply the Sturm rule to sequence (3) in the interval (−∞, +∞). As Dk (λ) = (−λ)k + . . . , then when λ changes from −∞ to +∞ the sequence (3) acquires exactly m sign changes, and therefore, by Sturm’s rule, 3◦ . All roots of the polynomial Dm (λ) are real and distinct. 4◦ . When λ passes through a root of Dm (λ), the product Dm Dm−1 reverses sign from + to −. Combining 4◦ with the Sturm rule, we obtain: 5◦ . The number of roots of the polynomial Dm (λ) on the interval (α, β) (α < β) is equal to the increase in the number of sign changes in the sequence (3) as λ changes from α to β. From 3◦ and 4◦ also follows 1 Here and from now on, when speaking of passing through a certain value, we shall mean a transition from smaller values to larger ones. 2 See also Grave [19], pp. 391 ff.

1. JACOBI MATRICES

69

6◦ . Between each two adjacent roots of the polynomial Dm (λ) lies exactly one root of the polynomial Dm−1 (λ) (m = 2, 3, . . . , n). We denote by λ1 < λ2 < · · · < λn all distinct eigenvalues of the matrix J, in other words, all the roots of Dn (λ). Let us show that: 7◦ . The sequence (4) Dn−1 (λ), Dn−2 (λ), . . . , D0 has j − 1 sign changes when λ = λj . Since the roots of the polynomial Dn (λ) separate the roots of the polynomial Dn−1 (λ), the interval (−∞, λj ) contains j − 1 roots of the polynomial Dn−1 (λ), located respectively in the intervals (λ1 , λ2 ), . . . , (λj−1 , λj ). Consequently, the increment in the number of sign changes in the sequence (4) should be equal to j − 1 as λ varies from −∞ to λj , But when λ = −∞ there are no sign changes in the sequence (4). This proves our statement. 2. Let us show how to calculate the coordinates of an eigenvector u = (u1 , u2 , . . . , un ), corresponding to an eigenvalue λ (Dn (λ) = 0). Let us write the vector equation Ju − λu = 0 in terms of coordinates ⎫ (a1 − λ)u1 − b1 u2 =0,⎪ ⎪ ⎪ −c1 u1 + (a2 − λ)u2 − b2 u3 =0,⎬ . (5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .⎪ ⎪ ⎪ ⎭ −cn−1 un−1 + (an − λ)un =0 Since the determinant Dn (λ) = 0 for the considered value of λ, and consequently, by 6◦ , Dn−1 (λ) = 0, it follows that the first n − 1 equations of (5) are linearly independent, and the last one follows from the first n − 1. Let us consider the first n − 1 equations, for any value of λ: − ck−1 uk−1 + (ak − λ)uk − bk uk+1 = 0 (k = 1, 2, . . . , n − 1; c0 = u0 = 0). Making the change of variables v1 = u1 ,

vk = b1 b2 . . . bk−1 uk

(k = 2, 3, . . . , n),

we obtain vk+1 = (ak − λ)vk − bk−1 ck−1 vk−1

(k = 1, 2, . . . , n − 1; v0 = 0).

(6)

Since the recurrence formula (6) from which the vk are determined coincides with (2) which serves to determine Dk−1 (λ), and, as can be readily verified, v1 = CD0 (λ),

v2 = CD1 (λ) (C = const = 0),

then vk = CDk−1 (λ) (k = 1, 2 . . . , n). Hence

−1 −1 uk = Cb−1 (7) 1 b2 . . . bk−1 Dk−1 (λ) (k = 1, 2, . . . , n). Letting now λ assume the value λj (j = 1, 2, . . . , n), we obtain for the k-th coordinate ukj of the j-th eigenvector uj the following expression: −1 ukj = Cj b−1 1 . . . bk−1 Dk−1 (λj ) (k = 1, 2, . . . , n; j = 1, 2, . . . , n).

(8)

70

II. OSCILLATORY MATRICES

Definition 1. We shall call a Jacobi matrix J normal if bk > 0, ck > 0 (k = 1, 2, . . . , n − 1). In most problems we deal with normal matrices. From (8), 3◦ and 7◦ we obtain the following theorem: Theorem 1. Normal Jacobi matrices have the following properties: 1◦ All eigenvalues are real and simple. 2◦ The sequence of coordinates of the j-th eigenvector has exactly j − 1 sign changes. We note that property 1◦ follows very easily also from the general theory of symmetric matrices. Indeed, since in the expression for Dn (λ) the quantities bk and ck (k = 1, 2, . . . , n − 1) enter only in the form of the products √bk ck (k = 1, 2, . . . , n−1), then replacing in the matrix J the numbers bk and ck by bk ck (k = 1, 2, . . . , n − 1), symmetrizes the matrix J without changing its characteristic equation. It follows that all eigenvalues are real. On the other hand, in view of the recurrent character of the system (5), to each eigenvalue corresponds exactly one eigenvector up to a scalar multiple. Consequently, the eigenvalues of the symmetrized matrix (see proposition 3◦ , Section 4, Chapter I), and hence of the given matrix, are simple roots of the polynomial Dn (λ). However, property 2◦ cannot be obtained from the general results of the theory of symmetric matrices: it depends on the special structure of the matrix J. 3. For further study of the spectral properties of normal matrices let us introduce the concept of the u(λ)-line of the matrix J. Definition 2. Let u = (u1 , u2 , . . . , un ) be a vector. Then the u-line is the piecewise linear curve in the plane with Cartesian coordinates X, Y , whose vertices Pk have coordinates xk = k, yk = uk

(k = 1, 2, . . . , n).

Obviously, for all points of intersection of a u-line with the X axis in the interval (1, n) to be crossing points, it is necessary and sufficient that whenever a coordinate uk (1 < k < n) of u is zero, the two adjacent coordinates uk−1 and uk+1 have opposite signs. Definition 3. Let u = (u1 , . . . , un ) be a vector. The points where the u-line crosses the X axis will be called the nodes of the u-line, or the nodes of the u-vector. Fixing in equation (7) an arbitrary value of the constant C (putting, for example, C = 1), we obtain a vector u(λ) = (u1 (λ), . . . , un (λ)) whose coordinates are functions of λ: u1 (λ) ≡ 1,

−1 −1 uk (λ) = b−1 1 b2 . . . bk−1 Dk−1 (λ) (k = 2, 3, . . . , n).

To this vector, according to Definition 2, will correspond a u(λ)-line, whose equation can be written in the following form: y(x; λ) = (k − x)uk−1 (λ) + (x − k + 1)uk (λ) for k − 1 ≤ x ≤ k

(k = 2, 3, . . . , n).

1. JACOBI MATRICES

71

We will investigate the shape of the u(λ)-line and the behavior of its nodes as λ varies. By the property 2◦ of the polynomials Dk (λ) (k = 0, 1, . . . , n−1), each common point of a u(λ)-line with the X axis in the interval (1, n) is a node of the u(λ)-line. By Theorem 1, a u(λ)-line with λ = λj has exactly j − 1 nodes. In general, for arbitrary λ by 5◦ , the number of nodes of the u(λ)-line is equal to the number of roots of the polynomial Dn−1 (λ), √ in the interval (−∞, λ). If we symmetrize the matrix J by replacing bk and ck by bk ck > 0 (k = 1, 2, . . . , n − 1), then the role of uk (λ) will be played by the quantities  −1 −1 −1 −1 −1 uk (λ) = C b−1 1 b2 . . . bk−1 c1 c2 . . . ck−1 Dk−1 (λ)  (9) b1 . . . bk−1 = uk (λ) (k = 1, 2, . . . , n). c1 . . . ck−1 Corresponding to the symmetrized matrix will be its own u(λ)-line P 1 . . . P n , which differs from the u(λ)-line P1 . . . Pn . However it follows from (9) that: 1) The number of nodes of these lines is always the same; 2) if one of these lines has a node between the points x = k − 1 and x = k, then the other line also has a node between these points; 3) if a variation of λ causes the nodes of one line to shift to the left, the same will occur for the second line, etc. Therefore, in proving the theorems that are to follow, we can assume without loss of generality that the matrix J is symmetric: bk = ck

(k = 0, 1, . . . , n − 1),

and that consequently the quantities uk (λ) (k = 0, 1, . . . , n) are related by the equations −bk−1 uk−1 (λ) + ak uk (λ) − bk uk+1 (λ) = λuk (λ) (10) (k = 0, 1, . . . , n − 1). Let us derive from these equations an identity which is fundamental for what is to come. For this purpose we replace λ in (10) by μ and eliminate from the resultant equations the coefficient ak ; we obtain bk−1 [uk−1 (λ)uk (μ) − uk−1 (μ)uk (λ)] −bk [uk (λ)uk+1 (μ) − uk (μ)uk+1 (λ)] = (μ − λ)uk (λ)uk (μ). Setting k equal to p, p + 1, . . . , q (1 ≤ p ≤ q < n − 1), and adding the resulting equations, we obtain: bp−1 [up−1 (λ)up (μ) − up−1 (μ)up (λ)] q  (μ − λ)uk (λ)uk (μ). −bq [uq (λ)uq+1 (μ) − uq (μ)uq+1 (λ)] =

(11)

k=p

In particular, when p = 1 the formula simplifies: −bq [uq (λ)uq+1 (μ) − uq (μ)uq+1 (λ)] q  (μ − λ)uk (λ)uk (μ) (1 ≤ q < n − 1). = k=1

With the aid of these formulas it is easy to prove the following theorems:

(12)

72

II. OSCILLATORY MATRICES

Theorem 2. If λ < μ, then between any two nodes of the u(λ)-line there is at least one node of the u(μ)-line. Proof. Let α, β (α < β) be two adjacent nodes of the u(λ)-line and p − 1 ≤ α < p,

q < β ≤ q + 1 (p ≤ q).

Consequently

 (p − α)up−1 (λ) + (α − p + 1)up (λ) = 0, (q + 1 − β)uq (λ) + (β − q)uq+1 (λ) = 0 and y(x; λ) = 0 for α < x < β. For the sake of definiteness, we assume that y(x; λ) > 0

for

α < x < β,

(13)

(14)

and consequently up (λ) > 0,

up+1 (λ) > 0, . . . , uq (λ) > 0.

(15)

Let us assume now, to the contrary to what we wish to prove, that y(x; μ) = 0

for

α < x < β.

Without loss of generality, we can assume that y(x; μ) > 0

for

α < x < β,

(16)

because in the opposite case we would consider −y(x; μ) instead of y(x; μ), replacing ui (μ) by −ui (μ); in this case the formula (11), with which we start, would remain in force. It follows from (16) that  (p − α)up−1 (μ) + (α − p + 1)up (μ) ≥ 0, (17) (q + 1 − β)uq (μ) + (β − q)uq+1 (μ) ≥ 0 and up (μ) > 0,

up+1 (μ) > 0,

...,

uq (μ) > 0.

(18)

Eliminating the quantities α and β from (13) and (17) we obtain up−1 (λ)up (μ) − up−1 (μ)up (λ) ≤ 0, uq (λ)uq+1 (μ) − uq (μ)uq+1 (λ) ≥ 0. Consequently, the left hand side of (11) is non-positive. At the same time, the right hand side of (11) is positive by (15) and (18). We have arrived at a contradiction, and this proves the theorem. Remark. In proving that at least one node of the u(μ)-line is located between α and β, we used only the fact that α and β are two zeros of the u(λ)-line; i.e., y(α; λ) = y(β; λ) = 0. Therefore this fact remains also true in that case, when α is a node of the u(λ)-line, and β = n and un (λ) = y(n; λ) = 0. This remark will be used later to prove Theorem 4. Theorem 3. As λ increases, the nodes of the u(λ)-line shift to the left.

1. JACOBI MATRICES

73

Proof. We denote by (0 λ at least one of the nodes α1 (μ) < α2 (μ) < . . . lies between the two adjacent nodes αk (λ) and αk+1 (λ), then to prove the inequalities αk (μ) < αk (λ) for λ < μ (k = 1, 2, . . . ) it is enough to prove the first of these. Using a proof by contradiction, we assume that α1 (μ) ≥ α1 (λ) (λ < μ). Let q < α1 (λ) ≤ q + 1 (1 ≤ q < n − 1). Since u1 (λ) ≡ 1 identically, then u1 (λ) > 0, u2 (λ) > 0, . . . , uq−1 (λ) > 0, uq (λ) > 0, (q + 1 − α)uq (λ) + (α1 − q)uq+1 (λ) = 0, u1 (μ) > 0, u2 (μ) > 0, . . . , uq−1 (μ) > 0, uq (μ) > 0, (q + 1 − α1 )uq (μ) + (α1 − q)uq+1 (μ) ≥ 0. Eliminating the quantity α1 from the relations that contain α1 , we obtain uq (λ)uq+1 (μ) − uq (μ)uq+1 (λ) ≥ 0. Consequently, in (12) the left hand side is non-positive, whereas the right hand side is positive. We arrive at a contradiction, thus proving the theorem. Theorem 4. The nodes of two successive eigenvectors alternate. Proof. By Theorem 1, the eigenvector uj , or in other words, the u(λj )-line has j − 1 nodes: α1 (λj ) < α2 (λj ) < · · · < αj−1 (λj )

(j = 2, 3, . . . , n).

By Theorem 3, α1 (λj+1 ) < α1 (λj ). Therefore, if we can prove that αj−1 (λj ) < αj (λj+1 ),

(19)

then, taking Theorem 2 into consideration, we obtain the required inequalities: α1 (λj+1 ) < α1 (λj ) < α2 (λj+1 ) < α2 (λj ) < · · · < αj−1 (λj ) < αj (λj+1 ). To prove inequality (19) we choose an arbitrary positive bn and put −1 −1 un+1 (λ) = b−1 1 b2 . . . bn Dn (λ).

(20)

Let us extend our u(λ)-line with one extra link Pn Pn+1 , where Pn+1 has coordinates xn+1 = n + 1,

yn+1 = un+1 (λ).

By (20), equation (10) now holds also for k = n. Then all the above results are applicable to the extended u(λ)-line P1 P2 . . . Pn Pn+1 . In particular, applying to the extended u(λj )-line the remarks after Theorem 2 (bearing in mind that

74

II. OSCILLATORY MATRICES

un+1 (λj ) = 0), we can verify that between the node aj−1 (λj ) and the point n + 1 there is at least one node of the extended u(λj+1 )-line. But since un+1 (λj+1 ) = 0, the extended u(λj+1 )-line has the same nodes α1 (λj+1 ) < α2 (λj+1 ) < · · · < αj (λj+1 ) as the initial u(λj+1 )-line. Hence αj−1 (λj ) < αj (λj+1 ) < n + 1. This proves the theorem. In addition to those properties which were stated in Theorems 1 and 4, the eigenvectors uj (j = 1, 2, . . . , n) of a normal Jacobi matrix satisfy several other remarkable oscillatory properties, which we shall obtain later together with the properties already established here, from more general considerations. In the derivation of the preceding theorems we made use of symmetrizability of the matrix J. However, a more refined analysis of the nature of oscillatory properties of matrices shows that the presence of these properties is not connected with symmetrizability or symmetry of the matrices. In the following sections we proceed to a study of a general type of nonsymmetric matrices, which have the basic oscillatory properties of normal Jacobi matrices. 2. Oscillatory matrices 1. We start with definitions. Definition 4. A matrix A = aik n1 will be called totally non-negative (or respectively totally positive) if all its minors of any order are non-negative (or respectively positive):   i1 < i2 < · · · < ip i1 i2 . . . ip A ≥ 0 (resp. > 0) for 1 ≤ ≤n k1 k2 . . . kp k1 < k2 < · · · < kp (p = 1, 2, . . . , n). Let us notice the simplest properties of totally non-negative matrices. 1◦ A product of two totally non-negative matrices is totally non-negative. 2◦ A product of a totally positive matrix by a nonsingular totally non-negative matrix is a totally positive matrix. We can verify these statements by recalling the expression for the minors of the product of two matrices in terms of the minors of the factors (see Sec. 1 of Chap. I): if C = AB, then   i1 i2 . . . ip C k1 k2 . . . kp      α1 α2 . . . αp i1 i2 . . . ip B A = α1 α2 . . . αp k1 k2 . . . kp α1 0)

(23)

is totally positive. As in the preceding example, it is enough to show that |A| > 0. But |A| = e−σ(α1 +···+αn +β1 +···+βn ) |e2σαi βk |n1 . 2

2

2

2

The determinant |e2σαi βk |n1 is positive, for it is obtained from the determinant of the matrix (21) by replacing ai with e2σαi and αk with βk . Let us put in (23) αi = βi = i (i = 1, 2, . . . , n) and denote the resulting totally positive matrix by Fσ : 2 Fσ = e−σ(i−k) n1 . (24) We note one property of the matrix Fσ , which we will use in the future: the diagonal elements of the matrix Fσ are equal to one, and all other elements tend to zero as σ → +∞. Therefore lim Fσ = E,

σ→+∞

where E is the unit matrix. 3. According to the well-known Cauchy formula4    n (xi − xk ) (yi − yk )  1  i ir+1 , the first two rows of the minor L kr . . . kp are proportional. Therefore this minor, and thus the minor (30) vanish. From a) and b) it follows that: c) A single-pair matrix L = lik n1 with elements  ψi χk (i ≤ k) , (33) lik = ψk χi (i ≥ k) where all the numbers ψ1 , . . . , ψn , χ1 , . . . , χn are non-zero, is totally non-negative if and only if all the numbers ψ1 , ψ2 , . . . , ψn , χ1 , χ2 , . . . , χn have the same sign and ψ1 ψ2 ψn ≤ ≤ ··· ≤ . χ1 χ2 χn

(34)

The rank r of the matrix L is equal to the number of < signs in (34) plus one. It follows from c), incidentally, that the matrix Gσ = e−σ|i−k| n1

(σ > 0)

is totally non-negative and nonsingular. 6. Let us consider a Jacobi matrix J = aik n1 : aik = 0

for |i − k| > 1.

Putting aii = ai (i = 1, 2, . . . , n), ak,k+1 = bk , ak+1,k = ck (k = 1, 2, . . . , n − 1),

(35)

80

II. OSCILLATORY MATRICES

we can write J in the form

  0 0   a1 b1 0 . . .   0 0   c1 a2 b2 . . .   0 0   0 c2 a 3 . . . J = .  ...........................     ...........................    0 0 0 . . . cn−1 an

(36)

Let us establish the following formula for the minors of the matrix J: a) If i1 < i2 < · · · < ip 1≤ ≤n k1 < k2 < · · · < kp and i1 = k1 , i2 = k2 , . . . , iν1 = kν1 ; iν1 +1 = kν1 +1 , . . . , iν2 = kν2 ; iν2 +1 = kν2 +1 , . . . , iν3 = kν3 ; . . . , then 6

 =J

i1 k1

... ...

  i1 i2 . . . ip J k1 k2 . . . kp       iν1 +1 iν2 iν2 +1 iν1 J ...J J kν1 kν1 +1 kν2 kν2 +1

To prove this formula iν = kν we have  i1 i2 . . . J k1 k2 . . .  i1 i2 . . . J k1 k2 . . .

... ...

iν3 kν3

(37)



(38) ...

it is enough to show that under the conditions (37) and ip kp ip kp



 =J



 =J

i1 k1

... ...

i1 k1

... ...

  iν+1 J kν+1   iν iν−1 J kν−1 kν iν kν

... ...

ip kp

... ...

ip kp

 (39)  (40)

Obviously it is enough to prove the first of these equalities. If iν < kν , then by (35) we have aiλ kμ = 0 (λ = 1, 2, . . . , ν; μ = ν + 1, . . . , p). If iν > kν , then aiλ kμ = 0 (λ = ν + 1, . . . , p; μ = 1, 2, . . . , ν). In either case the Laplace expansion of the determinant   i1 i2 . . . ip J k1 k2 . . . kp with respect to the first ν columns yields (39). Thus (38) is proved. From (38) it follows that:   i1 i2 . . . ip for which b) Every non-principal minor J k1 k2 . . . kp |iν − kν | ≤ 1  6 If

i1 = k1 , we must put ν1 = 0 and J

(ν = 1, 2, . . . , p) i1 k1

... ...

iν 1 kν1

 = 1 in (38).

(41)

3. EXAMPLES

81

can be represented by means of (38) in the form of a product of some principal minors and some of the numbers b and c. If (41) is not satisfied; i.e., if at least for one ν we have |iν − kν | > 1 (1 ≤ ν ≤ p), then

 J

i1 k1

i2 k2

... ...

ip kp

 = 0.

The following is a corollary from b): c) In order for a Jacobi matrix (36) to be totally non-negative it is necessary and sufficient that all the numbers b, c and the principal minors of this matrix be non-negative. To a totally non-negative matrix J corresponds a sign-regular matrix J ∗ , and vice versa. In this case J ∗ is obtained from J by reversing the signs of the numbers b and c. The principal minors of the matrix J ∗ are the same as those of J, since the numbers b and c enter in these minors only in the form of products bk ck (see the preceding section). Therefore, it follows from (c) that: d) In order for a Jacobi matrix (36) to be sign-regular, it is necessary and sufficient that all the numbers b and c be non-positive and the principal minors non-negative. In particular, a normal Jacobi matrix (see Sec. 1 of this Chapter, Definition 1) is sign-regular if and only if all its principal minors are non-negative. In the case of a nonsingular matrix J, it is possible to simplify criteria c) and d). Since the principal minors of the symmetrized matrix Js (see Sec. 1) and of the given J are the same, it follows from the non-negativity of the minors of J that the quadratic form Js (x, x) is non-negative. But since in addition the rank of J and consequently also of Js is equal to n, Js (x, x) is positive. Hence, all principal minors are positive. Conversely, if it is known that the successive principal minors of J are positive, it follows that the form Js (x, x) is positive, and consequently all principal minors of Js , and thus of J, are positive. Therefore: e) In order for a nonsingular Jacobi matrix to be totally non-negative, it is necessary and sufficient that all elements b and c be non-negative and that the successive principal minors be positive:    a1 b1 0       a1 b1   > 0,  c1 a2 b2  > 0, . . . , . a1 > 0,  (42)   c1 a2   0 c2 a 3  f ) In order for a nonsingular Jacobi matrix J to be sign-regular, it is necessary and sufficient that all the elements b and c be non-positive, and all the successive principal minors (42) be positive. In particular, a nonsingular normal Jacobi matrix is sign-regular if and only if all its successive principal minors are positive. Let J be a nonsingular Jacobi matrix and let L = lik n1 = J −1 ; i.e.   1 ... k − 1 k + 1 ... n J 1 ... i − 1 i + 1 ... n   . lik = (−1)i+k 1 2 ... n J 1 2 ... n We then conclude from (38) that

82

lik

II. OSCILLATORY MATRICES

1) When l ≤ k  1 ... J 1 ... = (−1)i+k  J = (−1)i+k

lik

1 ... 1 ...

2) When i > k  1 ... J 1 ... = (−1)i+k  J = (−1)i+k

1 ... 1 ...

    k + 1 ... k−1 i J ...J k + 1 ... k i+1   1 2 ... n J 1 2 ... n    i−1 k + 1 ... n bi bi+1 . . . bk−1 J i−1 k + 1 ... n   1 2 ... n J 1 2 ... n i−1 i−1

  J

    i + 1 ... i k+1 J ...J J i + 1 ... i−1 k   1 2 ... n J 1 2 ... n    k−1 i + 1 ... n ck ck+1 . . . ci−1 J k−1 i + 1 ... n   . 1 2 ... n J 1 2 ... n k−1 k−1





n n

n n





In particular, if J is a symmetric matrix (bi = ci for i = 1, 2, . . . , n − 1) and bi = 0 (i = 1, 2, . . . , n − 1), then putting ⎫   1 ... i − 1 ⎪ bi bi+1 . . . bn−1 ⎪ (−1)i J ⎪ ⎪ 1 ... i − 1 ⎪ ⎪ ⎪ ψi = ⎬ |J| , (43)   ⎪ i + 1 ... n ⎪ ⎪ (−1)i J ⎪ ⎪ i + 1 ... n ⎪ ⎪ ⎭ χi = bi bi+1 . . . bn−1 we can write

 lik =

ψi χk ψk χi

(i ≤ k) . (i > k)

Thus, g) If J is a nonsingular symmetric Jacobi matrix (36) with non-vanishing values of b, then the inverse matrix J −1 = L = lik n1 is a single-pair matrix  ψi χk (i ≤ k), lik = ψk χi (i > k) with non-vanishing numbers ψi and χi (i = 1, 2, . . . , n) which are determined by (43). It can be shown that, conversely, if L is a nonsingular single-pair matrix with non-vanishing numbers ψi and χi (i = 1, 2, . . . , n), then the inverse matrix L−1 is a symmetric Jacobi matrix with non-vanishing values of b. For this it is enough to use equations (29) and (30).

4. PERRON’S THEOREM

83

4. Perron’s theorem In the present section we shall establish Perron’s theorem [40] on the maximum eigenvalue and the corresponding eigenvector of a matrix with positive elements. This theorem will serve as a basis for the study of eigenvalues and eigenvectors of oscillatory matrices. For a symmetric matrix, Perron’s theorem can be proved much simpler (in contrast to the general case) by starting with “energy” considerations (the maximal property of the first eigenvalue of a quadratic form). Since one encounters in applications (Chap. III) only symmetric oscillatory matrices, we shall also give a proof for the case of symmetric matrices7 along with the general proof of Perron’s theorem, due to Frobenius [14a,b]. Theorem 5 (Perron). If all the elements of a matrix A = aik n1 are positive, then there exists a positive simple 8 eigenvalue ρ, which exceeds the moduli of all the other eigenvalues. To this “maximum” eigenvalue there corresponds an eigenvector with positive coordinates. First proof (for symmetric matrices). We denote by ρ the largest eigenvalue of a symmetric matrix A; i.e., the largest root of the equation |A − λE| = 0. Then according to Theorem 13 of Sec. 10 of Chap. I (in which we put C = E): A(x, x) ρ = max (x, x)

 n  A(x, x) = aik xi xk ;

(x, x) =

i,k=1

n 

 x2i

(44)

i=1

and this maximum is attained on any eigenvector z = (z1 , z2 , . . . , zn ) of the matrix A corresponding to the eigenvalue ρ:   A(z, z) Az = ρz z = 0, ρ = . (z, z) Let us introduce a vector u = (u1 , u2 , . . . , un ), where ui = |zi | (i = 1, 2, . . . , n). Since by agreement the quadratic form A(x, x) has positive coefficients, then A(z, z) ≤ A(u, u),

(45)

and at the same time (u, u) = (z, z). By (44), equality must hold in (45) and consequently, the maximum of interest to us is attained also on the vector u: ρ=

A(u, u) . (u, u)

Consequently (see Sec. 10, Chap. 1), Au = ρu; that is,

n 

aik uk = ρui

(i = 1, 2, . . . , n).

k=1 7 The reader interested principally in mechanical applications of oscillatory matrices may restrict himself to this proof. 8 i.e., which is a simple root of the characteristic equation.

84

II. OSCILLATORY MATRICES

Since all terms in the left hand side of each of these equations are non-negative and at least one of them is positive, we have (u = 0), then ρ > 0,

ui > 0 (i = 1, 2, . . . , n).

(46)

On the other hand, aik zi zk ≤ aik ui uk and

n 

aik zi zk =

i,k=1

(i, k = 1, 2, . . . , n), n 

aik ui uk .

(47)

(48)

i,k=1

This implies that we have equality in each of the relations in (47); i.e., all the zi have the same sign and z = ±u. (49) We have thus proved (see (46) and (49)) that for an arbitrary eigenvector z, corresponding to the eigenvalue ρ, all coordinates are different from zero and have the same sign. Consequently, there is no orthogonal pair among these eigenvectors, which implies that ρ is a simple eigenvalue. Corresponding to the eigenvalue ρ is an eigenvector u with positive coordinates. It remains to show that the greatest eigenvalue ρ exceeds the absolute value of any negative eigenvalue λ of the matrix A, provided, naturally, such negative eigenvalue exists. We denote by w = (w1 , w2 , . . . , wn ) an eigenvector corresponding to the eigenvalue λ . Since (u, w) = 0, the vector w has coordinates of different signs. Denoting w = (|w1 |, |w2 |, . . . , |wn |), we obtain: |λ | =

A(w , w ) |A(w, w)| < ≤ ρ, (w, w) (w , w )

q.e.d. Second proof (for the general case). We denote by Aik (λ) the cofactor of the element λδik − aik or the matrix λE − A = λδik − aik n1 . As soon as we prove the existence of a maximum eigenvalue ρ, then for the second part of Perron’s theorem it will be sufficient to prove this: Aik (ρ) > 0

(i, k = 1, 2, . . . , n).

Indeed, in this case, putting, for example, u1 = A11 (ρ), u2 = A12 (ρ), . . . , un = A1n (ρ), we have

n 

(ρδik − aik )uk = 0

(i = 1, 2, . . . , n),

k=1

and consequently u = (u1 , . . . , un ) will be an eigenvector with positive coordinates corresponding to the eigenvalue ρ. We claim that furthermore Aik (λ) > 0 for

λ ≥ ρ.

(50)

4. PERRON’S THEOREM

85

Let us prove Perron’s theorem together with the additional statement (50), by induction from n − 1 to n. For n = 1 we can consider the theorem to be true. Let us put Dm (λ) = |λδik − aik |m 1 (m = 1, 2, . . . , n). Expanding Dn (λ) with respect to the last row and the last column, we obtain n−1 

Dn (λ) = (λ − ann )Dn−1 (λ) −

(n−1)

Aik

(λ)ain ank ,

(51)

i,k=1 (n−1)

where Aik (λ) denotes the cofactor of the element λδik − aik in the determinant Dn−1 (λ). Assuming the theorem (together with (50)) proved for matrices of order < n, we denote by ρm the maximal eigenvalue of the truncated matrix Am = aik m 1 (m = 1, 2, . . . , n − 1). From (51) we obtain with λ = ρn−1 : Dn (ρn−1 ) = −

n−1 

(n−1)

Aik

(ρn−1 )ain ank < 0.

i,k=1

On the other hand, we have lim Dn (λ) = lim (λn + . . . ) = +∞,

λ→∞

λ→∞

and consequently, the equation Dn (λ) = 0 has a positive root in the interval (ρn−1 , ∞). The greatest positive root of the equation Dn (λ) = 0 will be denoted by ρn . Thus, ρn > ρn−1 . Analogously ρn−1 > ρn−2 > . . . ; consequently, ρn > ρm (m < n). The number ρm is the “maximal” root of the principal minor Dm (λ) in the determinant Dn (λ). By rearranging the rows and respectively the columns in the determinant Dn (λ), we can make any of its principal minors of order m < n play the role of the minor Dm (λ). Consequently ρn is greater than the largest root of any principal minor of the determinant Dn (λ). Since in addition the coefficient of the highest degree in the expansion of any principal minor in powers of λ in the determinant Dn (λ) is equal to one, all principal minors in Dn (λ) are positive when λ ≥ ρn . In particular, Aii (λ) > 0

for λ ≥ ρn

Hence Dn (ρn ) =

n 

(i = 1, 2, . . . , n).

Aii (ρn ) > 0;

k=1

i. e., ρn is a simple root of the characteristic equation Dn (λ) = 0. Let us consider now Aik (λ) for i = k. Expanding Aik (λ) with respect to the row and the column which are numbered respectively k and i in Dn (λ), we obtain:  Aik (λ) = aki C(λ) + Cpq (λ)api aqk (p, q = i, k); (52) p,q

Here C(λ) is the principal minor of order (n − 2) which is obtained from Dn (λ) by crossing out the row and column numbered i and k, and Cpq (λ) (p, q = i, k) denotes the co-factor of the element λδpq − apq in C(λ). As Perron’s theorem (together with

86

II. OSCILLATORY MATRICES

the supplementary statement (50)) is applicable to the matrix of the determinant C(λ) and the maximal root of the equation C(λ) = 0 is less than ρn , we see that C(λ) > 0 and

Cpq (λ) > 0 for λ ≥ ρn .

Consequently, by (52) Aik (λ) > 0 for λ ≥ ρn . To complete the proof of the theorem it remains to show that ρn > |λ0 |, where λ0 is any eigenvalue of the matrix A = aik n1 different from ρn . Writing ui = A1i (ρn ) (i = 1, 2, . . . , n) as before, we have n 

aik uk = ρn ui

(i = 1, 2, . . . , n).

(53)

k=1

Let v = (v1 , v2 , . . . , vn ) be any eigenvector of the transposed matrix A corresponding to the eigenvalue λ0 : n 

aik vi = λ0 vk

(k = 1, . . . , n).

(54)

i=1

Then

n 

aik |vi | ≥ |λ0 ||vk |

(k = 1, . . . , n).

(55)

i=1

From (53) and (55) we find that ρn

n  i=1

ui |vi | =

n  i,k=1

aik uk |vi | ≥ |λ0 |

n 

uk |vk |.

k=1

Therefore ρn ≥ |λ0 |. Equality occurs if and only if it occurs in every one of the relations (55), and consequently it occurs if and only if all non-vanishing complex numbers vi (i = 1, 2, . . . , n) have the same argument. But in this case, by (54), λ0 > 0, thus λ0 = ρn , which contradicts the assumption. Thus, ρn > |λ0 |. This proves Perron’s theorem completely. 5. Eigenvalues and eigenvectors of an oscillatory matrix In the present section we shall establish several important properties of eigenvalues and eigenvectors of an oscillatory matrix. Let us first introduce certain notions and notations. Let u1 , u2 , . . . , un (56) be a sequence of real numbers. If some of the terms of this sequence are zero, we can assign them arbitrarily chosen signs. We can then calculate the number of sign changes in the sequence (56). This number will change depending on our choice of signs for the zero terms of the sequence (56). The greatest and smallest values of this number will be called the maximal or respectively the minimal number of sign changes in the sequence (56) and we shall denote them by Su+ and Su− , respectively.

5. EIGENVALUES AND EIGENVECTORS OF AN OSCILLATORY MATRIX

87

In the case when Su+ = Su− , we speak of an exact number of sign changes in the sequence (56), and denote it simply by Su . Obviously this case can occur if and only if the following two conditions are satisfied: 1) u1 un = 0, and 2) if certain ui vanish (1 < i < n), then ui−1 ui+1 < 0. We note furthermore that Su is equal to the number of sign changes in the sequence (56) after all zero terms are removed. We note also that if all ui = 0 (i = 1, 2, . . . , n), then Su− = 0 and Su+ = n − 1; if among the terms ui (i = 1, 2, . . . , n) there are k (0 ≤ k < n) zero terms, then Su− ≤ n − k − 1 and Su+ ≥ k. We can now state the fundamental theorem on the spectral properties of an oscillatory matrix. Theorem 6. 1◦ An oscillatory matrix A = aik n1 always has only simple eigenvalues and they are all positive: λ1 > λ2 > · · · > λn > 0.

(57)

2◦ If uk = (u1k , u2k , . . . , unk ) is an eigenvector of an oscillatory matrix A corresponding to the eigenvalue λk , the k-th in magnitude among all eigenvalues, (k = 1, 2, . . . , n), then for every sequence of coefficients cp , cp+1 , . . . , cq , (1 ≤ p ≤ q  c2i > 0), the number of sign changes in the coordinates of the vector q ≤ n, i=p

u = cp up + cp+1 up+1 + · · · + cq uq lies between p − 1 and q − 1. More precisely, p − 1 ≤ Su− ≤ Su+ ≤ q − 1.

(58)

In particular, among the coordinates of the vector uk (k = 1, 2, . . . , n) there are exactly k − 1 sign changes, that is Suk = k − 1

(k = 1, 2, . . . , n).

(59)

3◦ The nodes of two successive eigenvectors uk and uk+1 (k = 1, 2, 3, . . . , n − 1) alternate. Proof. Let us first prove the theorem for the case when A is a totally positive matrix. Proof of 1◦ . If A is a totally positive matrix, then for any q ≤ n, all elements of the q-th associated matrix Aq (see Sec. 13, Chap. I) are positive, because all minors of q-th order of the matrix A are positive. Therefore Perron’s theorem applies to the matrix Aq . On the other hand, according to Kronecker’s theorem (Sec. 13, Chap. I) the eigenvalues of Aq are all possible products of q eigenvalues of A. If the eigenvalues of A are enumerated in the order of decrease of their absolute values, |λ1 | ≥ |λ2 | ≥ · · · ≥ |λn |, then the eigenvalue of Aq with the largest absolute value will be the product λ1 λ2 . . . λq . Therefore Perron’s theorem applied to the associated matrix Aq (q = 1, 2, . . . , n) yields immediately: λ1 λ2 . . . λq > 0 (q = 1, 2, . . . , n), λ1 λ2 . . . λq > |λ1 λ2 . . . λq−1 λq+1 | (q = 1, 2, . . . , n − 1).

88

II. OSCILLATORY MATRICES

From the first inequality it follows that λq > 0 (q = 1, . . . , n), and from the second inequality, λq > λq+1

(q = 1, 2, . . . , n − 1).

These are the inequalities (57). Proof of 2◦ . We proved that a totally positive matrix cannot have multiple eigenvalues. Therefore the eigenvector uk = (u1k , u2k , . . . , unk ) corresponding to the k-th eigenvalue λk is determined uniquely, up to a factor. Let us show that these factors can be chosen such that all minors   i1 i2 . . . iq (1 ≤ i1 < i2 < · · · < in ≤ n, q = 1, 2, . . . , n) U (60) 1 2 ... q of the fundamental matrix U = uik 1 be positive. Indeed, the minors (60), for a fixed q, are the coordinates of the eigenvector of the associated matrix Aq corresponding to the largest eigenvalue λ1 λ2 . . . λq (4◦ , Sec. 13, Chap. I). Therefore, by Perron’s theorem, all minors (60) are nonzero, and for any q ≤ n they have all have the same sign εq . If we multiply the vectors u1 , u2 , . . . , un by the factors ε1 , ε2 /ε1 , . . . , εn /εn−1 , we obtain the required inequalities:   i1 i2 . . . iq > 0 (1 ≤ i1 < i2 < · · · < iq ≤ n, q = 1, 2, . . . , n). (61) U 1 2 ... q Suppose that for an integer q we have   q q  c k uk c2k > 0 . u= k=1

(62)

k=1

Let us prove that (61) implies Su+ ≤ q − 1.

(63)

≥ q. We can then choose q+1 coordinates We assume the contrary; i.e., that ui1 , ui2 , . . . , uiq+1 of the vector u such that Su+

uiα uiα+1 ≤ 0 (α = 1, 2, . . . , q).

(64)

Here the ui1 , ui2 , . . . , uiq cannot simultaneously vanish, for in this case the numq  c2k > 0) would satisfy a system of homogeneous bers ck (k = 1, 2, . . . , 2, . . . , q, k=1

equations

q 

ck uia k = 0 (α = 1, 2, . . . , q)

k=1

with non-vanishing determinant (60). Let us now consider the determinant that is known to vanish:   ui 1 q ui 1   ui 1 1 . . .    ..........................    = 0.  ..........................    uiq+1 1 . . . uiq+1 q uiq+1

5. EIGENVALUES AND EIGENVECTORS OF AN OSCILLATORY MATRIX

89

Expanding this determinant with respect to the last column, we obtain   q+1  i1 . . . iα−1 iα+1 . . . iq+1 q+α+1 = 0. (−1) ui α U 1 ....................... q α=1

But such an equation is impossible, for by (61) and (64), all non-vanishing terms in the left hand side (and these are known to exist) have the same sign. We thus proved that (63) follows from (61) and (62). In order to complete the proof of 2◦ , let us consider a vector u = cp up + cp+1 up+1 + · · · + cq uq . We have proved that Su+ ≤ q − 1. Let us now put (see Sec. 2) B = A−1 = bik n1 ,

C = B ∗ = b∗ik  = (−1)i+k bik n1 .

Then C is a totally positive matrix (see 5◦ , Sec. 2). On the other hand, from Auk = λk uk follows

k Buk = λ−1 k u

(k = 1, 2, . . . , n) (k = 1, 2, . . . , n),

or, written out in full n 

bij ujk = λ−1 k uik

(i, k = 1, 2, . . . , n).

(65)

j=1

To each vector u = (u1 , u2 , . . . , un ) we assign a vector u∗ = (u∗1 , u∗2 , . . . , u∗n ), where u∗i = (−1)i ui (i = 1, 2, . . . , n). It then follows from (65) that n 

∗ b∗ij u∗jk = λ−1 k uik

(u∗ik = (−1)i+k uik ; i, k = 1, 2, . . . , n),

j=1

or

k∗ Cuk∗ = λ−1 (k = 1, 2, . . . , n). k u ∗ ∗ ∗ Thus, un , un−1 , . . . , u1 is a complete system of eigenvectors of a totally nonnegative matrix C. The corresponding eigenvalues are arranged in decreasing order: −1 −1 λ−1 n > λn−1 > · · · > λ1 . Therefore, since u∗ = cq u∗q + · · · + cp u∗p , we can write for the vector u∗ a relation which is analogous to (63) (here the role of the number q will be played by the number n − p + 1):

Su+∗ ≤ n − p.

(66)

But, as can be readily seen, we always have Su− + Su+∗ = n − 1.

(67)

It follows from (66) and (67) that Su− ≥ p − 1.

(68)

90

II. OSCILLATORY MATRICES

Combining (63) and (68), we obtain (58). This proves proposition 2◦ completely. Proof of 3◦ . Let us recall that according to Definitions 2 and 3 of Sec. 1 of this Chapter, the nodes of a vector u are the nodes of the corresponding u-line. In addition, since according to 2◦ , Su−k = Su+k , Su−k+1 = Su+k+1 ,

(69)

then all zeros of the uk -line and of the uk+1 -line are nodes. In proving proposition 3◦ we use only of the following facts, which follow from ◦ 2 : (70) Suk = k − 1, Suk+1 = k and for a vector u = cuk + duk+1 (71) 2 2 with arbitrary c and d (c + d > 0), we have k − 1 ≤ Su− ≤ Su+ ≤ k.

(72)

Let us now assume that between some two successive nodes α and β of the uk+1 -line there are no nodes of the uk -line. Then on (α, β), the functions uk (x) and uk+1 (x) (the ordinates of the uk and uk+1 lines) are different from zero and have constant signs. Without loss of generality, we can assume uk (x) > 0, uk+1 (x) > 0 (α < x < β).

(73)

Let us put for convenience d = −1 in (71) and consider the function u(x) = cuk (x) − uk+1 (x). First we show that uk (α) = 0 and uk (β) = 0. Indeed, let, for example, uk (α) = 0. Then for every c we have u(α) = 0. We choose γ, satisfying α < γ < min{β, [α + 1]}, where [α + 1] is the integer part of α + 1, and put c = uk+1 (γ)/uk (γ). Then we also get u(γ) = 0; i.e., a segment of the u-line lies on the x axis. This means that two successive coordinates of the vector u vanish. So Su+ − Su− ≥ 2, which contradicts (72). This shows that the uk -line and the uk+1 -line do not have common nodes. Let us notice now that for sufficiently large c we have by (73): u(x) > 0 (α ≤ x ≤ β). Let us decrease c to a value c0 , at which u(x) first vanishes at least at one point γ ∈ [α, β]. It can be readily seen that c0 > 0. In addition, the function u0 (x) = c0 uk (x) − uk+1 (x) ≥ 0 (α ≤ x ≤ β) does not vanish when x = α and when x = β; consequently, the root γ of this function lies in (α, β). Thus, the broken line y = u0 (x) lies on one side of the X-axis, and has a common point P (γ, 0) with the X axis (α < γ < β). It may happen that an entire link of our broken line y = u0 (x) lies on the x-axis. But then, as we have already explained, Su+0 − Su−0 ≥ 2, which contradicts (72). On the other hand, if the broken line has only one vertex P (γ, 0) on the X-axis, then the two links that meet at this vertex are located on the same side

6. A FUNDAMENTAL DETERMINANTAL INEQUALITY

91

of the X axis, thus among the coordinates of the vector u0 there is one coordinate that vanishes, and the two adjacent coordinates have the same sign. In this case, as can be readily seen, again Su+0 − Su−0 ≥ 2, and again we have a contradiction with (72). Thus, between two successive nodes of the uk+1 -line lies at least one node of the uk -line. But, according to (70), the uk -line has k − 1 nodes and the uk+1 -line has k nodes. This implies that between two successive nodes of the uk+1 line lies exactly one node of the uk -line, thus the nodes of the uk -line and of the uk+1 -line alternate. So the theorem is proved for the case when A is a totally positive matrix. If A is an arbitrary oscillatory matrix with exponent κ, then the matrices Aκ and Aκ+1 are totally positive (see Sec. 2); the eigenvalues of these matrices will be respectively λκ1 , λκ2 , . . . , λκn and λκ+1 , λκ+1 , . . . , λκ+1 n . 1 2 ◦ Applying proposition 1 to the totally positive matrices Aκ and Aκ+1 , we obtain λκ1 > λκ2 > · · · > λκn > 0 and

λκ+1 > λκ+1 > · · · > λκ+1 > 0. n 1 2

From this we readily obtain (57). The matrices A and Aκ have the same eigenvectors, wherein the vector uk corresponds to the k-th eigenvalues λk and λκk of the matrices A and Aκ . Therefore, propositions 2◦ and 3◦ hold also for the eigenvectors of the oscillatory matrix A. This proves the theorem completely. 6. A fundamental determinantal inequality Let us consider a minor of a matrix A = aik n1     i1 < i2 < · · · < ip i1 i2 . . . ip 1≤ A ≤n . k1 k2 . . . kp k1 < k2 < · · · < kp If (74) is a principal minor, we have

p 

(74)

|iν − kν | = 0.

ν=1

Definition 7. A minor (74) will be called almost principal, if of the differences i1 − k1 , . . . , ip − kp only one is not zero. Each element aik (i = k) is an almost principal minor of order one. For p > 1, every almost-principal minor has the form   i . . . iα−1 i iα+1 . . . iβ−1 iβ iβ+1 . . . ip A 1 , (i = k) (75) i1 . . . iα−1 iα+1 iα+2 . . . iβ k iβ+1 . . . ip and it is obtained from a principal minor of order p − 1 by adding one row and one column with arbitrary distinct subscripts. Theorem 7. If in a matrix A = aik n1 all principal and almost principal minors are non-negative, then for every p < n, the following inequality holds       p + 1 ... n 1 ... p 1 2 ... n . (76) A ≤A A p + 1 ... n 1 ... p 1 2 ... n

92

II. OSCILLATORY MATRICES

Proof. For n = 2, this is immediate:    a11 a12     a21 a22  = a11 a22 − a12 a21 ≤ a11 a22 ,

(77)

because by assumption a12 a21 ≥ 0. We prove the theorem by induction with respect to n. If n > 2, then one of the numbers p or n − p is greater than 1; without loss of generality, we can assume that p > 1.9 We consider two cases. 1. First suppose that   1 2 ... p − 1 = 0. A 1 2 ... p − 1 We introduce the matrix D = dik np , whose elements are defined by equations   1 ... p − 1 i (i, k = p, . . . , n). (78) dik = A 1 ... p − 1 k By Sylvester’s identity  i1 i2 . . . D k1 k2 . . .  1 2 ... =A 1 2 ...

(Sec. 2 of Chap. I),  iq kq q−1  1 2 p−1 A 1 2 p−1

p−1 p−1

... ...

i1 k1

i2 k2

... ...

iq kq

 ,

the principal minors of D are positive, and the almost-principal ones are nonnegative. Since by the induction assumption the theorem is true for matrices of order < n, then using Sylvester’s identity twice, we can write     p p + 1 ... n p + 1 ... n  D  dpp D p p + 1 ... n p + 1 ... n 1 ... n =  A ≤ n−p n−p  1 ... n 1 ... p− 1 1 ... p− 1 A A 1 ... p− 1 1 ... p− 1     1 ... p − 1 p + 1 ... n 1 ... p (79) A A 1 ... p − 1 p + 1 ... n 1 ... p   = 1 ... p − 1 A 1 ... p − 1     p + 1 ... n 1 ... p . A ≤A p + 1 ... n 1 ... p 2. Suppose now that

 A

1 2 ... 1 2 ...

p−1 p−1

 = 0.

Then either a11 = 0 or there exists q < p − 1 such that    1 2 ... 1 2 ... q = 0 and A A 1 2 ... 1 2 ... q 9 Otherwise

q+1 q+1

 = 0.

(80)

case we would renumber all the rows and column of the matrix A in reverse order.

6. A FUNDAMENTAL DETERMINANTAL INEQUALITY

If a11 = 0 then it follows from      a11 a1k  1 i   ≥ 0, = A 1 k ai1 aik 

93

ai1 a1k ≥ 0 (i, k = 2, 3, . . . , n),

that ai1 a1k = 0 (i, k = 2, 3, . . . , n). So ai1 = 0

(i = 2, 3, . . . , n)

or a1k = 0 (k = 2, 3, . . . , n). As a11 = 0 in any of these cases  1 2 ... A 1 2 ... If a11 = 0 then we have (80). We put   1 ... q i bik = A 1 ... q k

n n

 = 0.

(i, k = q + 1, . . . , n).

(81)

(82)

Then all principal and almost principal minors of the matrix B = bik pq+1 are non-negative; this follows from Sylvester’s Identity g−1      i1 i2 . . . ig 1 ... q 1 . . . q i1 i2 . . . ig B =A A 1 ... q k1 k2 . . . kg 1 . . . q k1 k2 . . . kg (83) and from the assumption of the theorem. Using the fact that the matrix B satisfies (76) (because its order is less than n), and that bq+1,q+1 = 0 (which follows from (80) and (81)), we conclude that     q + 2 ... n q + 1 ... n =0 ≤ bq+1,q+1 B B q + 2 ... n q + 1 ... n 

thus B

q + 1 ... q + 1 ...

n n

 = 0.

Applying Sylvester’s Identity (83) again, we obtain (81). Thus in the considered case the left hand side of (76) is zero. As he right hand side is non-negative, inequality (76) follows. This proves the theorem. Theorem 8. If A = aik n1 is a totally non-negative matrix, then for any p < n the inequality (76) holds:       p + 1 ... n 1 ... p 1 2 ... n . A ≤A A p + 1 ... n 1 ... p 1 2 ... n Here equality holds only in the two obvious cases: 1) One of the factors in the right hand side vanishes; 2) All elements aik (i = 1, . . . , p; k = p + 1, . . . , n) or all elements aik (i = p + 1, . . . , n; k = 1, . . . , p) vanish.

94

II. OSCILLATORY MATRICES

Proof. Every totally non-negative matrix satisfies the conditions of the previous theorem, so (76) holds. To prove the second part of the theorem, we need following lemma, which is of independent interest. Lemma 1. If A = aik n1 is a totally non-negative matrix, and   1 2 ... p − 1 q = 0 (1 < p < q ≤ n), A 1 2 ... p − 1 p 

then either A

1 1

2 2

... ...

p−1 p−1

p p

(84)

 = 0,

or aqk = 0 Proof. Assume that



A In view of (76) we have  1 2 ... A 1 2 ...

1 2 ... 1 2 ...

p−1 p p−1 p

consequently,

(k = 1, 2, . . . , p). p−1 p p−1 p



 ≤A

 > 0.

1 2 ... 1 2 ...

p−1 p−1

 app ;

 1 2 ... p − 1 > 0. 1 2 ... p − 1 It follows from (84) and (85) that in the matrix   ... a1p   a11    ...................     ap−1,1 . . . ap−1,p     ...................    ... aqp   aq1    ...................    an1 ... anp 

A

(85)

the first p−1 rows are linearly independent, and the q-th row is a linear combination of these p − 1 rows: aqk =

p−1 

λν aνk

(k = 1, 2, . . . , p).

ν=1

Let us show that all λν = 0. Indeed, for any ν (1 ≤ ν ≤ p − 1)   1 2 ... ν − 1 ν + 1 ... p − 1 q A 1 2 ... ν − 1 ν ... p − 2 p − 1   1 2 ... p − 1 , = (−1)p−ν−1 λν A 1 2 ... p − 1   1 2 ... ν − 1 ν + 1 ... p q A 1 2 ... ν − 1 ν ... p − 1 p   1 2 ... p . = (−1)p−ν λν A 1 2 ... p

(86)

(87)

6. A FUNDAMENTAL DETERMINANTAL INEQUALITY

95

Since all minors of A are non-negative and the minors in the right hand sides of the inequalities (86) and (87) are positive, we have the simultaneous inequalities (−1)p−ν−1 λν ≥ 0 and

(−1)p−ν λν ≥ 0;

i.e. λν = 0 (ν = 1, 2, . . . , p − 1). This proves the lemma. We now proceed with the proof of the second part of Theorem 8; i.e., with the proof that in (76) equality occurs only in the cases 1) and 2) of Theorem 8. When n = 2 this is obvious, for in this case    a11 a12     a21 a22  = a11 a22 and consequently a12 a21 = 0. We prove our statement by induction, assuming that it is correct for matrices of order < n. As in the proof of Theorem 7, without loss of generality, we can assume that p > 1. Let us show now that for every n, if       p + 1 ... n 1 ... p 1 ... n = 0, (88) A =A A p + 1 ... n 1 ... p 1 ... n then case 2) holds. We notice that (88) implies   1 ... p − 1 = 0, A 1 ... p − 1 for in the opposite case the inequality    1 1 ... p ≤A A 1 1 ... p

... ...

p−1 p−1

(89)  app

would not be satisfied (a particular case of the inequality (76) applied to the totally non-negative matrix aik p1 ). We can now write the chain of inequalities (79) from the proof of Theorem 7. Since equality should hold in all steps of this chain then, taking (88) into account, we get     p p + 1 ... n p + 1 ... n = dpp D = 0. (90) D p p + 1 ... n p + 1 ... n   p p + 1 ... n Applying the second part of Theorem 8 to the determinant D p p + 1 ... n we obtain one of the following two systems of equations:

Let, for example, dip = A



dip = 0

(i = p + 1, . . . , n),

(91)

dpk = 0

(k = p + 1, . . . , n).

(92)

1 ... 1 ...

p−1 p−1

i p

 =0

(i = p + 1, . . . , n).

96

II. OSCILLATORY MATRICES

Then, taking (88) into account, we find using Lemma 1: aik = 0 (i = p + 1, . . . , n; k = 1, 2, . . . , p). Exactly in the same manner, we obtain from (92) the following equations aik = 0 (i = 1, 2, . . . , p; k = p + 1, . . . , n). This proves Theorem 8 completely. In Theorem 8, inequality (76) was established under the assumption that certain minors of the matrix A = aik n1 are non-negative. In Sec. 8 of Chap. I this inequality was established on the assumption that A = aik n1 is a symmetric matrix, n  composed of the coefficients of a positive quadratic form aik xi xk (generalized i,k=1

Hadamard’s inequality). These two conditions do not imply each other, and they are both contained as special cases in the following more general conditions for validity of (76). Theorem 9. If in a matrix A = aik n1 , all principal minors are positive, and the products of any two almost principal minors symmetric with respect to the main diagonal are non-negative, then inequality (76) holds for every p < n. To prove this theorem we suggest that the reader repeat verbatim the proof of Theorem 7 for the case when the principal minors are positive (part 1 of the proof). Remark. Let us show how Theorem 10 of Sec. 8, Chap. I and Theorem 8 of the present section follow from Theorem 9. n  If A is a symmetric matrix and the quadratic form aik xi xk is positive, i,k=1

then it is easy to see that the conditions 1) and 2) of Theorem 9 are satisfied and n  (76) therefore holds. If the form aik xi xk is non-negative, it can be considered i,k=1

as a limit of a sequence of positive forms, and by passing to the limit one obtains (76). If A is a totally non-negative matrix with vanishing determinant then (76) holds because its left hand side is zero while the right hand side is non-negative. If A is a non-singular totally non-negative matrix, one can approximate it by totally positive matrices: A = lim Fσ AFσ , σ→∞

where Fσ is defined in (24) (Section 3 of this Chapter). Totally positive matrices satisfy the assumptions of Theorem 9, so (76) holds for them. This implies that (76) also holds for non-singular totally non-negative matrices. Corollary. If the conditions of one of the the following inequality holds  1 ... p p + 1 ... q q + 1 ... A 1 ... p p + 1 ... q q + 1 ...    p+1 1 ... p p + 1 ... q A ≤A p+1 1 ... p p + 1 ... q

Theorems 7, 8, or 9 are satisfied, n n



... ...





A

p+1 p+1

... ...

q q

q q

q+1 q+1

... ...

n n



(93) .

7. CRITERION FOR A MATRIX TO BE OSCILLATORY

97

Proof. In the proof we can assume that |A| = 0, since when |A| = 0 the inequality (93) is obvious.  = Aik  adjoint to A, so that We consider the matrix A   1 ... i − 1 i + 1 ... n (i, k = 1, 2, . . . , n). (94) Aik = A 1 ... k − 1 k + 1 ... n Using the formula for the minors of the adjoint matrix ((4) of Chap. I), we obtain:        h1 . . . hp = |A|p−1 A h1 . . . hn−p , A (95) l1 . . . lp l1 . . . ln−p  are the complementary systems of indices for where h1 , . . . , hn−p and l1 , . . . , ln−p the index systems h1 , . . . , hp and l1 , . . . , lp , respectively. From this we conclude  the conditions of the Theorem 7, 8, or 9 are satisfied just as that for the matrix A for the matrix A. These conditions are satisfied also for the matrix Aik  Therefore   1 ... p A 1 ... p

(i, k = 1, . . . , p; q + 1, . . . , n).

q + 1 ... q + 1 ...

n n



 ≤A



1 1

... ...

p p



 A



q + 1 ... q + 1 ...

n n

 .

(96)

 by (95) in terms of the minors of A and dividing Expressing here the minors of A both sides of the inequality by |A|n+p−q−2 , we obtain (93). 7. Criterion for a matrix to be oscillatory In this section we shall establish a criterion for a totally non-negative matrix to be an oscillatory matrix. This criterion plays a fundamental role in establishing that specific matrices encountered in applications are oscillatory. 1. We shall first prove several auxiliary propositions. 1◦ If A = aik n1 is an oscillatory matrix, then every truncated matrix aik qp (1 ≤ p < q ≤ n) is also oscillatory. Proof. It is obviously enough to prove this proposition for the matrix B = aik n2 . Let the matrix Aχ be totally positive; we shall show that B χ will also be totally positive; in other words, we shall show that for any two systems of indices i1 < i2 < · · · < ip ≤ n, k1 < k2 < · · · < kp

2≤ the minor

=

 

B  A

2≤α 0 (i = 1, 2, . . . , n − 1).

(115)

7. CRITERION FOR A MATRIX TO BE OSCILLATORY

103

Similarly we establish that ci+1,i > 0 (i = 1, 2, . . . , n − 1).

(116)

Using the criterion of the oscillatory character, we conclude from (114), (115), and (116) that C is an oscillatory matrix. 3. Let us apply the criterion of the oscillatory character to obtain certain important examples of oscillatory matrices. Let us consider a Jacobi matrix   0 0   a1 b1 0 0 . . .   0 0   c1 a2 b2 0 . . .   0 0   0 c2 a3 b3 . . . (117) J = .  ...............................     ...............................    0 0 0 0 . . . cn−1 an . Theorem 11. 1) In order for a Jacobi matrix (117) to be oscillatory, it is necessary and sufficient that all numbers b and c be positive, and that the successive principal minors be positive:    a1 b1 0       a1 b1   > 0,  c1 a2 b2  > 0, a1 > 0,  (118)    c1 a2  0 c2 a 3  2) The exponent of an oscillatory Jacobi matrix is always equal to n − 1, where n is the order of the matrix. Proof. 1. The necessity and sufficiency of these follow from the criterion of oscillatory character (Theorem 10) taking into account proposition e) of example 6 of Sec. 3. 2. According to 2◦ , the exponent of every oscillatory matrix is ≤ n − 1. On the other hand, putting J κ = hik n1 , we notice that when κ < n − 1, the element h1n = 0, and thus J κ is not a totally positive matrix. Consequently, the exponent of J is n − 1. Theorem 12. In order for a single-pair matrix L = lik n1 , where  ψi χk (i ≤ k) . lik = ψk χi (i ≥ k),

(119)

to be an oscillatory matrix, it is necessary and sufficient that the following conditions be satisfied: 1) The 2n numbers ψ1 , ψ2 , . . . , ψn , χ1 , χ2 , . . . , χn are different from zero and have the same sign; ψ1 ψ2 ψn 2) < < ··· < . χ1 χ2 χn Proof. The conditions are necessary. Indeed, if L is an oscillatory matrix, then all principal minors of L are positive. Therefore, lii = ψi χi > 0 (i = 1, 2, . . . , n),   1  lii ψi li,i+1  ψi+1 = − > 0. 2   l l lii χi+1 i+1,i i+1,i+1 χi+1 χi

(120) (121)

104

II. OSCILLATORY MATRICES

Since lik ≥ 0 (i, k = 1, 2, . . . , n), (119) and (120) imply that all the elements lik > 0 (i, k = 1, . . . , n) and from this we obtain condition 1). Condition 2) follows from (121). The conditions are sufficient. Indeed, if these conditions are satisfied, then by part c) of example 5 of Sec. 3, we conclude that L is a totally non-negative matrix. In addition, li+1,i = li,i+1 = ψi χi+1 > 0 and, by (29) (Sect. 3), we have for the minors of a single-pair matrix         χ χ2   χ2 χ3   χn−1 χn  1 2 ... n  ...  χ > 0. = ψ1  1 L 1 2 ... n ψ1 ψ2   ψ2 ψ3   ψn−1 ψn  n Thus, according to the criterion of oscillatory character, L is an oscillatory matrix. Let J be an oscillatory Jacobi matrix. Then the matrix M = (J ∗ )−1 is also oscillatory (see Sec. 2, 9◦ ). In the case of a symmetric matrix J, the matrix M is a single-pair matrix (see proposition g) of example 6, Sec. 3). In the general case, when J is not necessarily symmetric, the matrix M can be considered as some generalization of a single-pair matrix. We notice the following properties of the matrices J and M : 7◦ In an oscillatory Jacobi matrix J, those and only those minors   i1 i2 . . . ip J k1 k2 . . . kp do not vanish, for which |i1 − k1 | ≤ 1, |i2 − k2 | ≤ 1, . . . , |ip − kp | ≤ 1. ◦

(122)

∗ −1

8 In an oscillatory matrix M = (J ) those and only those minors   i1 i2 . . . ip M k1 k2 . . . kp do not vanish, for which i1 , k1 < i2 , k2 < · · · < ip , kp .

(123)

Proof of propositions 7◦ and 8◦ . As was established in Sec. 3 (example 6), a minor of the matrix J, for which relations (122) are not satisfied is equal to zero; and a minor whose indices satisfy (122) factors into a product of several principal minors and certain numbers b and c. Since in an oscillatory matrix, all principal minors and all numbers b and c are positive, proposition 7◦ follows. Proposition 8◦ follows from proposition 7◦ by the formula   l1 l2 . . . ln−p  J  j1 j2 . . . jn−p i1 i2 . . . ip   = M k1 k2 . . . kp 1 2 ... n J 1 2 ... n (see Sec. 2, Chap. I), where both systems of indices i1 < i2 < · · · < ip ; j1 < j2 < · · · < jn−p , and k1 < k2 < · · · < kp ; l1 < l2 < · · · < ln−p coincide with the

8. PROPERTIES OF THE CHARACTERISTIC DETERMINANT

105

system of indices 1, 2, . . . , n. We leave it for the reader to prove that if the systems of indices i, k satisfy (123), then the complementary systems of indices j, l satisfy (122). Propositions 7◦ and 8◦ are of certain interest, if one recalls that in an arbitrary oscillatory matrix, the minors which simultaneously satisfy conditions (122) and (123) (quasi-principal minors) are non-vanishing. 8. Properties of the characteristic determinant of an oscillatory matrix Let there be given an arbitrary oscillatory matrix A = aik n1 . Similarly to what has been done in Sec. 1 of this chapter for Jacobi matrices12 , we introduce a u(λ)-vector, whose coordinates u1 (λ), u2 (λ), . . . , un (λ) are the cofactors of the corresponding elements of the last row of the matrix λδik − aik n1 ; in particular un (λ) = |λδik − aik |n−1 . (124) 1 Evidently,

n 

(λδik − aik )uk (λ) = 0 (i = 1, 2, . . . , n − 1)

(125)

k=1

and

n 

(λδnk − ank )uk (λ) = D(λ),

(126)

k=1 aik |n1 .

where D(λ) = |λδik − When λ = λj (j = 1, 2, . . . , n), the vector u(λ) = (u1 (λ), . . . , un (λ)) becomes the eigenvector uj of the matrix A corresponding to the eigenvalue λj and therefore the sequence u1 (λj ), u2 (λj ), . . . , un (λj ) has exactly j − 1 sign changes. Let us prove the following theorem: Theorem 13. The sequence of polynomials un (λ), un−1 (λ), . . . , u1 (λ)

(127)

in the interval (0, ∞) has the three characteristic properties of ordinary Sturm sequences, namely: 1◦ The polynomial u1 (λ) has constant sign for 0 < λ < ∞. 2◦ If for some λ > 0, an inner term ui (λ) (1 < i < n) vanishes, then the adjacent terms ui−1 (λ) and ui+1 (λ) have opposite signs. 3◦ When λ passes through a root of un (λ) in the increasing direction, the product un (λ)un−1 (λ) changes sign from − to +. 12 In comparing the results of this section with the analogous results of Sec. 1, the reader will observe certain differences. These are due to the fact that if a normal Jacobi matrix J has positive eigenvalues, then according to Theorem 11 it is not the matrix J but the matrix J ∗ which is oscillatory.

106

II. OSCILLATORY MATRICES

Proof. To prove property 1◦ we notice that   a13 ... a1,n−1 a1n   a12   a23 ... a2,n−1 a2n   a22 − λ   a33 − λ . . . a3,n−1 a3n  u1 (λ) =  a32    .............................................    an−1,2 an−1,3 . . . an−1,n−1 − λ an−1,n   a23 ... a2,n−1 a2n   a22 − λ   a33 − λ . . . a3,n−1 a3n   a32   = (−1)n  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ,    an−1,2 an−1,3 . . . an−1,n−1 − λ an−1,n    a12 a13 ... a1,n−1 a1n and consequently

    a1i a1n    n−3  aii ain  λ i>1  a1k a1n  aik ain  λn−4 + . . . akk akn 

u1 (λ) = a1n λn−2 +    a1i  aii +  k>i>1  aki

All coefficients of the polynomial u1 (λ) are non-negative. In addition, the polynomial u1 (λ) is not identical zero, since from the fact that in the sequence (127) there are exactly j − 1 sign changes when λ = λj , it follows that u1 (λj ) = 0 (j = 1, 2, . . . , n). Thus, at least one of the coefficients of the polynomial u1 (λ) is positive and u1 (λ) > 0 when 0 < λ < ∞. Property 2◦ means that when un (λ) = 0, the vector u(λ) = (u1 (λ), . . . , un (λ)) has a well defined number of sign changes; i.e., + − Su(λ) = Su(λ) ,

(λ) = (u1 (λ), . . . , un−1 (λ)) has a well and when un (λ) = 0, the reduced vector u defined number of sign changes; i.e. − Su+ ˜(λ) = Su ˜(λ) .

Let us consider the following three cases: 1) The number λ = λ0 > 0 is such that un (λ0 ) = 0 and t=

D(λ0 ) > 0. un (λ0 )

(128)

We denote by At the matrix which is obtained from A by replacing the element ann by ann + t. Since t > 0 then     i1 < i2 < · · · < ip i1 . . . ip i1 . . . ip At ≥A for 1 ≤ ≤n k1 . . . kp k1 . . . kp k1 < k2 < · · · < kp (p = 1, . . . , n), and consequently, the matrix At is also an oscillatory matrix. From (128) it also follows that λ0 is an eigenvalue of the matrix At since |λE − At | = D(λ) − tun (λ).

8. PROPERTIES OF THE CHARACTERISTIC DETERMINANT

107

On the other hand, since the matrices At and A differ only by one element in the last row, it follows from (125) that the vector u(λ0 ) = (u1 (λ0 ), . . . , un (λ0 )) is an eigenvector of the matrix At , corresponding to the eigenvalue λ0 . By Theorem 6, + − Su(λ = Su(λ . 0) 0) 2) The number λ = λ0 > 0 is such that un (λ0 ) = 0 and τ=

−D(λ0 ) > 0. λ0 un (λ0 )

We denote by A(τ ) the matrix which is obtained from the matrix A by dividing all elements in the last row by 1 + τ . Obviously, A(τ ) is an oscillatory matrix. It is easily seen that (1 + τ )|λE − A(τ ) | = D(λ) + τ λun (λ), and consequently λ0 is an eigenvalue of the matrix A(τ ) . For the same reasons as in case 1), the vector u(λ0 ) is an eigenvector of the matrix A(τ ) and consequently, + − again Su(λ = Su(λ . 0) 0) Let us consider the last case. 3) The number λ = λ0 > 0 is such that un (λ0 ) = 0. It follows then from the relations (125) that the reduced vector u (λ0 ) = (u1 (λ0 ), . . . , un−1 (λ0 )) is an , and consequently, Su+ eigenvector of the reduced oscillatory matrix aik n−1 1 ˜(λ0 ) = − Su˜(λ0 ) . To prove property 3◦ , we use properties 1◦ , 2◦ from Sec. 1 of this Chapter. When λ varies between α and β, the number of sign changes in the sequence (127) (un (α) = 0, un (β) = 0, α < β) increases, by the difference between the number of roots of the function un (λ) in the interval (α, β) at which the product un (λ)un−1 (λ) changes sign from + to −, and the number of roots in the same interval, at which this product changes sign from − to +. For λ = λn , the sequence (127) has exactly n − 1 sign changes, and for λ = λ1 > λn it has no sign changes, and the polynomial un (λ) is of degree n − 1. This is possible only if all the roots of the polynomial un (λ) lie between λn and λ1 , and the product un−1 (λ)un (λ) changes sign from − to + when passing through each root of un (λ). This proves the theorem. Corollary. The number of zeros of un (λ) in an interval (α, β) (0 < α < β, un (α) = 0, un (β) = 0) is equal to the decrease in number of sign changes in the sequence (127) as λ varies from α to β. In particular, since for λ = λj+1 (j = 1, 2, . . . , n − 1) the sequence (127) has exactly j sign changes, and for λ = λj it has exactly j − 1 sign changes, un (λ) has one and only one root between λj+1 and λj . Thus the roots of the polynomials D(λ) and un (λ) alternate. Recalling the expression (124) for un (λ), and the fact that together with the matrix aij n1 all the matrices aij k1 (k = 1, 2, . . . , n) are also oscillatory, we arrive to the following theorem: Theorem 14. If a matrix A = aik n1 is oscillatory, then the roots every two adjacent polynomials in the sequence D1 (λ), D2 (λ), . . . , Dn (λ),

108

II. OSCILLATORY MATRICES

   a11 − λ . . . a1k   Dk (λ) =  . . . . . . . . . . . . . . . . . . . . . .   ak1 . . . akk − λ 

where

(k = 1, 2, . . . , n),

alternate.13 From the fact that the roots of the polynomials Dn (λ) and Dn−1 (λ) alternate, one deduces the following theorem: Theorem 15. If U = uik n1 is a fundamental matrix of an oscillatory matrix A = aik n1 , and V = (U  )−1 = vik n1 , then the first and last coordinates of each pair of vectors uj = (u1j , u2j , . . . , unj ) and v j = (v1j , v2j , . . . , vnj ) (j = 1, 2, . . . , n) have identical signs; i.e. u1j v1j > 0, unj vnj > 0

for

j = 1, 2, . . . , n.

(129)

Proof. By the relation A = U λi δik n1 U −1 we have A − λE = U (λi − λ)δik n1 U −1 , and consequently

 n     δik n −1   U = U  δik  V  . (A − λE)−1 = U   λi − λ   λi − λ  1 1

From this, equating the elements at the intersection of the last row and the last column, we find that n Dn−1 (λ)  unj vnj = . D(λ) λ −λ j=1 j Since, by virtue of the fact that the roots of the polynomials D(λ) = (−λ)n +. . . and Dn−1 (λ) = (−λ)n−1 + . . . alternate, the ratio of these polynomials Dn−1 (λ)/Dn (λ) changes from −∞ to +∞ as λ passes through a root λj (j = 1, 2, . . . , n), and thus all the products unj vnj (j = 1, 2, . . . , n) are positive. From symmetry considerations (or from the fact that the vectors uj and v j (j = 1, 2, . . . , n) have the same number of sign changes) it also follows that u1j v1j > 0 (j = 1, 2, . . . , n).

9. Eigenvalues of an oscillatory matrix as functions of its elements 1. Theorem 16. If λ1 and λn are respectively the greatest and the smallest eigenvalues of an oscillatory matrix A = aik n1 , then ∂λn ∂λ1 > 0, (−1)i+k >0 ∂aik ∂aik 13 It

(i, k = 1, 2, . . . , n).

is interesting to compare this theorem with Theorem 20 of Sec. 10, Chap. I.

9. EIGENVALUES OF AN OSCILLATORY MATRIX

109

Proof. As in the preceding section, we introduce the matrices U and V ; then V  AU = λi δik n1 ; hence λj =

n 

ars usj vrj

(j = 1, 2, . . . , n).

r,s=1

Differentiating both sides of this equation with respect to aik , we obtain n n   ∂λj ∂vrj ∂usj = ukj vij + ars usj + ars vrj . ∂aik ∂aik r,s=1 ∂aik r,s=1

By the relations n 

ars usj = λj urj ,

s=1

we find that

and

n 

ars vrj = λj vsj ,

r=1

n ∂λj ∂  = ukj vij + λj urj vrj = ukj vij , ∂aik ∂aik r=1

(130)

since from the equation U  V = E follows that: n 

urj vrj = 1.

(131)

r=1

According to Theorem 6, all numbers u11 , u21 , . . . , un1 are different from zero and have the same sign εu . The same can be said about v11 , v21 , . . . , vn1 , since the matrix V = (U  )−1 is fundamental for the oscillatory matrix A (see Sec. 3, Chap. I, p. 19). The common sign of the numbers v11 , v21 , . . . , vn1 will be denoted by εv . From (131) (with j = 1) we get εu = εv , so it follows from (130) that ∂λ1 = uk1 vi1 > 0. ∂aik On the other hand, by Theorem 6, Sec. 5, among the coordinates of the vectors un = (u1n , u2n , . . . , unn ) and v n = (v1n , v2n , . . . , vnn ) there are exactly n − 1 sign changes. Therefore sign ukn = (−1)k−1 sign u1n ,

and sign vin = (−1)i−1 sign v1n .

Hence, by (130) and Theorem 15 of Section 8, sign

∂λn = sign [(−1)i+k u1n v1n ] = (−1)i+k . ∂aik

This proves the theorem. Theorem 17. If A = aik n1 is an oscillatory matrix with eigenvalues λ1 > λ2 > · · · > λn , then ∂λj ∂λj ∂λj ∂λj > 0, > 0; (−1)j−1 > 0, (−1)j−1 >0 ∂a11 ∂ann ∂a1n ∂an1 (j = 1, 2, . . . , n).

110

II. OSCILLATORY MATRICES

Proof. The first inequality follows directly from (129) and (130). The second inequality becomes the first one if the numeration of the rows and the columns is reversed; with this the matrix remains oscillatory and its eigenvalues do not change. The third and fourth inequalities are obtained if one notices that by (130), ∂λj ∂λj = unj v1j , = u1j vnj ∂a1n ∂an1 and, furthermore,

⎫ sign vnj = (−1)j−1 sign v1j , ⎪ ⎬

sign unj = (−1)j−1 sign u1j , ⎪ ⎭ u1j v1j > 0

(j = 1, 2, . . . , n).

This proves the theorem. 2. Let us consider now a normal Jacobi symmetric matrix (see Sec. 1)   0 ... 0 0   a1 −b1   0 0   −b1 a2 −b2 . . .   −b2 a3 . . . 0 0   0 J = .  ..................................     ..................................    0 0 0 . . . −bn−1 an We denote by λ1 < λ2 < · · · < λn the eigenvalues of the matrix J and by U = ujk n1 its fundamental matrix. The matrix J of course is not oscillatory (in the case that J(x, x) is a positive form, a normal Jacobi matrix becomes an oscillatory matrix when −bk is replaced by bk (k = 1, 2, . . . , n − 1)); nevertheless, we can apply formula (130) to the matrix J, since in the derivation of this formula we used only the fact that the matrix has simple structure. Since for a symmetric matrix J the matrix U can be chosen orthogonal, we have V = (U  )−1 = U . Therefore (130) yields: ∂λj = u2ij ≥ 0 (j = 1, 2, . . . , n), ∂ai ∂λj = −ukj uk+1,j ∂bk

(j = 1, 2, . . . , n; k = 1, 2, . . . , n − 1).

(132) (133)

It follows from (132) that: 1◦ Each eigenvalue λj is a non-decreasing function of a1 , a2 , . . . , an .14 Let us consider the special particular cases of (133) for j = 1 and j = n. It was established in Sec. 1 that in the sequence of coordinates of the j-th (j = 1, 2, . . . , n) eigenvector (u1j , u2j , . . . , unj ) there are exactly j − 1 sign changes. Therefore uk1 uk+1,1 > 0 and

ukn uk+1,n < 0,

and thus, by (133), ∂λ1 < 0, ∂bk

and

∂λn > 0 (k = 1, 2, . . . , n − 1); ∂bk

14 By Theorem 15 of Chap. I this proposition is true for every symmetric matrix A = a n ; ik 1 i.e., ∂λj /∂akk ≥ 0 (j, k = 1, 2, . . . , n).

9. EIGENVALUES OF AN OSCILLATORY MATRIX

111

2◦ The smallest eigenvalue λ1 of a normal symmetric Jacobi matrix is a decreasing function of b1 , b2 , . . . , bn−1 . 3◦ The greatest eigenvalue λn of a normal symmetric Jacobi matrix is an increasing function of b1 , b2 , . . . , bn−1 . It follows from propositions 1◦ , 2◦ , and 3◦ that by replacing all the ai by their largest value (max ai ) or by the smallest value (min ai ), and proceeding analogously with the numbers bk , we obtain upper and lower bounds for λ1 and λn . In order to compute these bounds, we determine λ1 and λn for a matrix J0 , with a1 = a2 = · · · = an = a and b1 = b2 = · · · = bn−1 = b. For this purpose we write the equations for the coordinates of the j-th eigenvector −buk−1 + (a − λ0j )uk − buk+1 = 0 (k = 1, 2, . . . , n; u0 = un+1 = 0), where

λ0j

(134)

(j = 1, 2, . . . , n) are the eigenvalues of the matrix J0 . Putting

1 − λ0j = cos θj , 2b we reduce equation (134) to the following difference equation:

(135)

uk−1 − 2 cos θj uk + uk+1 = 0. The general solution of such an equation is given by the formula uk = A sin kθj + B cos kθj . Now we take into account the additional conditions u0 = un+1 = 0. From u0 = 0 follows B = 0, and since then A = 0, the condition un+1 = 0 implies sin(n + 1)θj = 0. Consequently, we can write θj =

πj n+1

(j = 1, 2, . . . , n).

From (135) we obtain πj (j = 1, 2, . . . , n). n+1 Using these expressions for the eigenvalues of the matrix J0 , we obtain from propositions 1◦ , 2◦ , and 3◦ the following estimates for the eigenvalues λ1 and λn of J: π π max b ≤ λ1 ≤ max a − 2 cos min b, min a − 2 cos n+1 n+1 π π min b ≤ λ1 ≤ max a + 2 cos max b. min a − 2 cos n+1 n+1 λ0j = a − 2b cos

http://dx.doi.org/10.1090/chel/345/04

CHAPTER III

Small Oscillations of Mechanical Systems with n Degrees of Freedom 1. Equations of small oscillations 1. Let S be a mechanical system with n degrees of freedom, capable of oscillating about its stable equilibrium position. Let q1 , . . . , qn be the generalized coordinates of the system S; without loss of generality, we assume that the coordinates are chosen such that in the equilibrium position is q1 = 0, . . . , qn = 0. According to the general assumptions usually made in investigating small oscillations of a system, we can assume that the kinetic energy T and the potential energy V are represented by quadratic forms with constant coefficients: n n 1  1  T = cik q˙i q˙k , V = aik qi qk , 2 2 i,k=1

i,k=1

where q˙i = dqi /dt (i = 1, 2, . . . , n) is the i-th generalized velocity. Since the kinetic energy of a moving system is positive, T is a positive quadratic form of q˙1 , . . . , q˙n . By virtue of the positiveness of the form T , there exists, by Theorem 11 of Chapter I, Sec. 9 a transformation of the coordinates qi (i = 1, 2, . . . , n) to certain new coordinates θi (i = 1, 2, . . . , n). qi =

n 

uik θk

(i = 1, 2, . . . , n),

(1)

k=1

which reduces both T and V to sums of squares1 : 1  ˙2 θ , 2 i=1 i n

T =

1 λi θi2 . 2 i=1 n

V =

(2)

1 In this theorem we spoke of simultaneous reduction of two forms A(x, x) and C(x, x) in the same set of variables x1 , x2 , . . . , xn ; on the other hand, in the forms T and V the variables are different. But if we subject the variables qi (i = 1, . . . , n) to the transformation (1) with constant coefficients uik (i, k = 1, 2, . . . , n), then the time derivatives q˙i (i = 1, 2, . . . , n) are subjected to the transformation n  q˙i = uik θk k=1

with the same matrix of coefficients uik n 1 , and therefore (2) is satisfied. 113

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III. SYSTEMS WITH n DEGREES OF FREEDOM

Here the real numbers λi (i = 1, 2, . . . , n) are the roots of the generalized characteristic equation |aik − λcik |n1 = 0, which may be written in greater detail in the following form:    a11 − λc11 a12 − λc12 . . . a1n − λc1n     a21 − λc21 a22 − λc22 . . . a2n − λc2n  (3)   = 0.  .........................................    an1 − λcn1 an2 − λcn2 . . . ann − λcnn As regards the matrix uik n1 , it is made up of the coordinates of the vectors ui = (u1i , . . . , uni ) (i = 1, 2, . . . , n) which, first of all, are solutions of the system of the equations ⎫ (a11 − λc11 )u1 + · · · + (a1n − λc1n )un = 0 ⎪ ⎬ (4) ......................................... ⎪ ⎭ (an1 − λcn1 )u1 + · · · + (ann − λcnn )un = 0 with λ = λi (i = 1, 2, . . . , n) and which, in addition, are so chosen and normalized that n  C(ui , uk ) = cμν uμi uνk = δik (i, k = 1, . . . , n). (5) μ,ν=1

We recall also that the defect d = n−r (where r is the rank) of the system (4) at λ = λi is equal to the multiplicity of λi as the root of the equation (3). Therefore, if λi is a simple root, the numbers uk (k = 1, . . . , n) are determined from the system (4) with λ = λi up to proportionality, and discarding that equation of (4) which is linearly dependent on the remaining ones, we find that u1 u2 un = = ··· = = γ, Δ1 Δ2 Δn where Δi (i = 1, 2, . . . , n) are the cofactors of the successive elements of the discarded row in the matrix ajk − λi cjk n1 . The coefficient γ is determined from the normalization condition n  C(u, u) = γ cμν Δμ Δν = 1. μ,ν=1

The coordinates θ1 , . . . , θn are called principal coordinates or normal 2 coordinates of this system. Setting up the Lagrange equations for the motion of our system in principal coordinates: d ∂T ∂V + =0 dt ∂ θ˙i ∂θi

(i = 1, 2, . . . , n),

2 Sometimes normal coordinates are defined in a somewhat broader manner, namely, as the coordinates θ1 , . . . , θn in which T and V assume the form

T =

n  i=1

Obviously, in this case λi =

ai ci

ci θ˙i2 , V =

n 

ai θi2 .

i=1

(i = 1, 2, . . . , n).

1. EQUATIONS OF SMALL OSCILLATIONS

115

we obtain a system of n differential equations θ¨i + λi θi = 0 (i = 1, . . . , n). If λi > 0, then the solution of the i-th equation has the term θi = Ai cos λi t + Bi sin λi t.

(6)

(7)

If λi < 0, then the ordinary sines and cosines are replaced in this formula by hyperbolic ones. On the other hand, if λi = 0, then the general solution of (6) has the form θi = Ai + Bi t. In the latter two cases motions with unbounded θi are possible for all initial data, no matter how small. Therefore, if, as assumed, the position of the system qi = 0 (i = 1, . . . , n) is a stable equilibrium, then λi > 0 (i = 1, 2, . . . , n); (8) i.e., the form V is positive. The last condition is not only necessary but also sufficient (for the stability of the equilibrium position qi = 0 (i = 1, 2, . . . , n)), for when it is satisfied, θi (i = 1, 2, . . . , n) is obtained from formulas (7), where 1 Ai = θi0 , Bi = √ θ˙i0 λi

(i = 1, 2, . . . , n),

(9)

and from these equalities and (7) it follows that for sufficiently small initially given θi0 and θ˙i0 (i = 1, 2, . . . , n) the coordinates θi (i = 1, 2, . . . , n) will remain as small as desired during the entire motion. Formula (7) can also be written as θi = Ri sin( λi t + αi ) (i = 1, 2, . . . , n), where Ri is the amplitude of the harmonic quantity θi (i = 1, 2, . . . , n) and αi is its initial phase; with this  Ai Ri = A2i + Bi2 , αi = arctan (i = 1, 2, . . . , n). Bi Inserting into (1) the just obtained expressions for θi (i = 1, 2, . . . , n) we obtain an expression for qi (i = 1, 2, . . . , n) as functions of time t. The solution thus obtained can be represented in a more compact form by changing to vector notation. Introducing the vector q = (q1 , q2 , . . . , qn ) we write the transformation (1) in abbreviated form as follows: q = θ1 u 1 + θ 2 u 2 + · · · + θ n u n ,

(10)

and consequently, finally: q=

n 

Ri sin( λi t + αi )ui .

(11)

i=1

We see that the vector q is the sum of the vectors q i = Ri sin( λi t + αi )ui (i = 1, 2, . . . , n).

(12)

This fact, expressed in words, is: the general free oscillation of the system is obtained by superposition of the harmonic oscillations of the system, taken with arbitrary phases and amplitudes.

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III. SYSTEMS WITH n DEGREES OF FREEDOM

Simultaneously we see that the numbers λi (i = 1, 2, . . . , n) which we have obtained earlier and the vectors ui (i = 1, 2, . . . , n) have the following mechanical meaning: √ The number λi (i = 1, 2, . . . , n) is the square of the frequency pi (pi = λi ) of the i-th harmonic oscillation (12) of the system, and the vector ui (i = 1, 2, . . . , n) specifies the “direction” of this oscillation. Let us show now how we can express the arbitrary integration constants Ai and Bi (i = 1, 2, . . . , n) and thus the quantities Ri and αi (i = 1, 2, . . . , n), directly in terms of the initial conditions q10 and q˙10 (i = 1, 2, . . . , n). For this purpose we form the C-products of both sides of equation (10) with the vector uν (ν = 1, 2, . . . , n); by (5) we obtain θν = C(uν , q) =

n 

cik uiν qk .

i,k=1

Consequently, taking (9) into consideration, we find that Aν = C(uν , q 0 )

1 and Bν = √ C(uν , q˙0 ) λi

(ν = 1, 2, . . . , n).

Thus, the problem of determining the motion of the system S has been completely solved. Before proceeding to establish certain general laws of mechanical oscillations, we note further that usually one arrives at the frequency equation (3) and the system of equations (4) in the following manner: We set up the Lagrange equations d ∂T ∂V + = 0 (i = 1, 2, . . . , n) dt ∂ q˙i ∂qi starting with the expressions for T and V in the original coordinates qi (i = 1, 2, . . . , n) and obtain n 

aik qk +

k=1

n 

cik q¨k = 0 (i = 1, 2, . . . , n).

(13)

k=1

Since we wish to compose the general solution of this differential system from particular harmonic solutions, we seek a solution q in the form q = u sin (pt + α); i.e., we put in (13) qi = ui sin(pt + α) (i = 1, 2, . . . , n). After this substitution and cancellation of sin(pt + α) we arrive at the equations n 

(aik − cik p2 )uk = 0

(i = 1, 2, . . . , n),

k=1

which coincide with (4) for λ = p2 . Equating the determinant of this system to 0, we obtain the frequency equation (3): |aik − cik p2 |n1 = 0. Historically this was the first method, and it was precisely in this way, starting with problems of mechanics, that mathematicians developed the theory of the

1. EQUATIONS OF SMALL OSCILLATIONS

117

characteristic equation, and generally, studied the properties of families of quadratic forms. 2. Let us arrange the frequencies pi of harmonic (or, as they also say, natural) oscillations in increasing order: p1 ≤ p 2 ≤ · · · ≤ p n . The natural oscillation with the lowest frequency pi is usually called the fundamental tone of the system, and the natural oscillations with frequencies p2 , p3 , . . . are called respectively the first, second, etc. overtones of the system. In Sec. 10 of Chap. I we established several minimax properties of the eigenvalues λi (i = 1, 2, . . . , n). These properties √ now acquire a simple intuitively obvious mechanical meaning for the numbers pi = λi ; we shall formulate them in the form of laws. Law I. As the inertia of the system increases while the stiffness remains unchanged, the frequencies of the system do not increase and at least one of them decreases. Here an increase in inertia of the system is meant to be a transition from our system with kinetic energy T = 12 cik qi qk to a system with greater kinetic energy   T  = 12 cik qi qk (so that the form T  − T is non-negative and is not identically 0). By conservation (respectively increase) of the stiffness of a system we mean the conservation (increase) of its potential energy. That this terminology is natural becomes clear if we recall that when the inertia mass of the bodies under consideration is increased, the kinetic energy of this system increases; on the other hand, if the stiffness of an elastic system or of its fastenings is increased, then for a given deformation its potential energy is increased. Law I is obviously a consequence of Theorem 16 of Chap. I. It can be refined by using Theorem 18 of Chap. I. From the same Theorem 16 follows Law II. As the stiffness of the system is increased without changing its inertia, the frequencies of the system do not decrease and at least one of them increases. Theorem 19 of Chap. I can now be stated as Law III. If k constraints are imposed on a system, the successive frequencies p1 ≤ p2 ≤ · · · ≤ pn−k of the constrained system are not less than the corresponding frequencies p1 ≤ p2 ≤ · · · ≤ pn−k and are not greater than the corresponding frequencies pk+1 ≤ pk+2 ≤ · · · ≤ pn of the free system. We do not specify here that the constraints are linear, because for small oscillations we neglect second and higher order terms in the coordinates, and therefore each constraint is considered as linear (in addition, it should be homogeneous, since it is satisfied at values qi = 0 (i = 1, 2, . . . , n)). The following law is a supplement to III and a translation of Theorem 15 of Chap. I into mechanical language. Law IV. The p-th overtone of an oscillating system S coincides with the highest of all the fundamental tones of all mechanical systems, obtained from the system S by imposing p constraints. In the next section we shall supplement these laws with several others, but now subject to certain limitations concerning the type of the forms for the kinetic and potential energies.

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III. SYSTEMS WITH n DEGREES OF FREEDOM

2. Oscillations of Sturm systems Definition 1. A mechanical system will be called a Sturm system 3 if the form T does not contain products of different generalized velocities, and the form V is a normal Jacobi form; i.e., T =

n 

ci q˙i2 ,

i=1

V =

n 

ai qi2 − 2

i=1

n−1 

bi qi qi+1

i=1

(bi > 0, i = 1, 2, . . . , n − 1). In this case, the Lagrange equations assume the form ci q¨i − bi−1 qi−1 + ai qi − bi qi+1 = 0

(i = 1, 2, . . . , n; b0 = bn = 0).

Consequently, if u = (u1 , . . . , un ) is the amplitude vector of a harmonic oscillation, q = u sin(pt + α), we obtain the equations: −bi−1 ui−1 + ai ui − bi ui+1 − ci p2 ui = 0 (i = 1, 2, . . . , n),

(14)

−γi−1 ui−1 + (αi − λ)ui − βi ui+1 = 0 (λ = p2 ; i = 1, 2, . . . , n),

(15)

or where

ai (i = 1, 2, . . . , n), ci bi (i = 1, 2, . . . , n), βi = ci bi (i = 0, 1, . . . , n − 1). γi = ci+1 Equating the determinant of the system (14) or (15) to 0, we obtain an equation for its frequencies; as we have seen, this turns out to be the characteristic equation of a Jacobi matrix. Recalling the results of Section 1 of Chapter II, related to Jacobi matrices, we can state the following theorem: αi =

Theorem 1. The frequencies of harmonic oscillations of a Sturm system are all distinct. Furthermore: 1◦ The amplitude vector of the fundamental tone has all coordinates of the same sign. 2◦ The amplitude vector of the k-th overtone (k = 1, 2, . . . , n − 1) has exactly k nodes. 3◦ The nodes of the amplitude vectors of two adjacent overtones alternate.4 . 3 Sturm, as has been discovered in manuscripts found after his death (see Bocher [7b]) was the first to investigate in detail oscillations of these systems. Furthermore, starting with the properties of these systems Sturm discovered his remarkable theorem of higher algebra. A study of the system of algebraic equations (15) is intimately related with an investigation of the same Sturm [53] on second-order differential equations, called the Sturm–Liouville equations. 4 Furthermore, by Theorem 2 of Chap. II, between each two nodes of the k-th overtone there is at least one node of any succeeding overtone.

2. OSCILLATIONS OF STURM SYSTEMS

119

We notice in addition that if we put   −b1 0 ... 0   a1 − λc1   a2 − λc2 −b2 ... 0   −b1   Δk (λ) =  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ,   −bk−1   .................................   0 0 . . . −bk−1 ak − λck (k = 1, 2, . . . , n), we obtain the following recurrence relation for the functions Δk (λ): Δk (λ) = (ak − λck )Δk−1 (λ) − b2k−1 Δk−2 (λ) (k = 2, 3, . . . , n;

Δ0 = 1).

(16)

With the aid of this relation we can compute the polynomial Δn (λ), the roots of which are the squares of the frequencies λi = p2i (i = 1, 2, . . . , n) of our system. The coordinates of an amplitude vector u = (u1 , . . . , un ) of a harmonic oscillation with frequency p (p = p1 , p2 , . . . , pn ) are found from the formulas −1 −1 2 uk = γb−1 1 b2 . . . bk−1 Δk−1 (p ) (k = 1, 2, . . . , n),

(17)

where γ is an arbitrary constant. These expressions for uk (k = 1, 2, . . . , n) may be verified by direct substitution in (14); here it is necessary to take into account the recurrence relation (16). Incidentally, formulas (16) and (17) follow also from formulas (2) and (7) of Section 1 of Chap. II. Let us consider certain specific problems, which lead to Sturm systems. 1. Vibrations of a thread with beads. We investigate transverse oscillations of a massless ideally flexible thread with n beads. This problem plays a unique role in the history of mechanics and mathematics. It was perhaps the first problem in the investigation of small oscillations of a system with n degrees of freedom. In connection with this problem, d’Alembert proposed his method of integrating systems of linear differential equations with constant coefficients. Starting with this problem, Daniel Bernoulli stated his remarkable hypothesis that the solution of the problem of free oscillations of a string can be represented in the form of a trigonometric series, which gave rise to a discussion between Euler, d’Alembert, Bernoulli and others on the nature of trigonometric series, a discussion which extended over several decades. Later on, Lagrange showed more rigorously, how it is possible, by passing to the limit, to obtain from the solution of a problem of an oscillating thread with beads, the solution of the problem of the oscillating string. Finally, as we have already noted (see footnote on page 160), this problem (and an analogous problem from the theory of heat conduction) was used by Sturm to guide his remarkable investigations in higher algebra and the theory of differential equations. Thus, assume that on a thread of unit length, stretched between its two stationary ends with tension σ, are placed n beads. We denote by m1 , m2 , . . . , mn the masses of these beads and by l0 , l1 , . . . , ln the lengths of the successive segments into which the thread is subdivided by the beads in the equilibrium position. We shall consider small transverse oscillations of the beads and we consequently neglect their shifts in the direction of the line of equilibrium of the thread. Choosing

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III. SYSTEMS WITH n DEGREES OF FREEDOM

some positive direction of the Y axis, perpendicular to the line of equilibrium of the thread, we denote by y1 , y2 , . . . , yn the displacements of the beads parallel to the Y axis. We then have the following expression for the kinetic energy: 1 mi y˙i2 . 2 i=1 n

T =

(18)

Assuming the tension of the thread to be constant (= σ) as the beads are displaced (this can be assumed in view of the smallness of these displacements) we can write V = σΔl, where Δl is the elongation of the thread due to the displacement of the beads. But, for specified displacements yi (i = 1, 2, . . . , n), the elongation of a segment li (i = 0, 1, . . . , n) is obviously given by the formula:    (yi+1 − yi )2 2 2 1+ − 1 = Δli = li + (yi+1 − yi ) − li = li li2 =

1 (yi+1 − yi )2 + . . . 2li

(i = 0, 1, . . . , n; y0 = yn+1 = 0).

Since, obviously, Δl = ΣΔli , then, up to small quantities of fourth order we have the following expression for V : V =

n σ1 (yi+1 − yi )2 2 i=0 li

(y0 = yn+1 = 0).

(19)

It is seen from (18) and (19) that a thread with beads is a Sturm system and consequently Theorem 1 can be applied to the oscillations of the thread. From this theorem it follows that in the case of harmonic oscillations of the thread with minimal frequency (i.e., at the fundamental tone), all parts of the thread are bent simultaneously either to one side or to the opposite side; in the case of harmonic oscillations producing the k-th overtone, the thread has k stationary nodes. If we recall furthermore how we defined the u-line of a vector u = (u1 , . . . , un ) and its nodes in Section 1 of Chap. II, it becomes clear that in statement 3◦ of Theorem 1, instead of the nodes of the amplitude vector we can speak of nodes of the thread itself (provided it is not in the zero phase). We can thus state that the nodes of adjacent overtones of the vibrating thread alternate and furthermore that between each two nodes of one overtone there lies at least one node of any succeeding overtone. We shall explain also later (see Section 10 of this Chapter) how the variable nodes of the oscillating thread with beads behave in complex oscillation which results from a superposition of several harmonic oscillations. If one of the ends of the thread is not secured, bearing a certain mass m0 ≥ 0, and can slide freely in a direction perpendicular to the line of equilibrium of the thread (we imagine for example that the left end of the thread is connected to a small ring of mass m0 , which slides freely on an ideally smooth wire threaded through it), we again have a Sturm system, for which now 1 T = mi y˙ i2 , 2 i=0 n

n σ1 V = (yi+1 − yi )2 2 i=0 li

(yn+1 = 0).

(20)

If we neglect the mass m0 (the mass of the small ring), then the form T is not positive anymore (one square is missing). But in this case, from the Lagrange

2. OSCILLATIONS OF STURM SYSTEMS

121

equation, d ∂T ∂V σ + = m0 y¨0 + (y0 − y1 ) = 0, dt ∂ y˙ 0 ∂y0 l1 in view of m0 = 0, we obtain y0 = y1 . We can therefore again consider our system as a Sturm system, but now with n (rather than with n + 1) degrees of freedom, discarding from formulas (20) for T and V the first (zero) terms. If both ends of the thread bear certain masses m0 ≥ 0 and mn+1 ≥ 0 and can slide freely in directions perpendicular to the line of equilibrium of the thread, we again have a Sturm system for which now T =

n+1 1 mi y˙i2 , 2 i=0

V =

n σ  (yi+1 − yi )2 , 2 i=0 li

(21)

but in this case the system is unstable (the thread, as a whole, can move inertially in one direction), and this is expressed mathematically by the fact that the form for V , while non-negative, is no longer positive (since now there are n + 2 variables and n + 1 squares, and V = 0 when y0 = y1 = · · · = yn = yn+1 = 0). In the formulation of Theorem 1, it is necessary to distinguish between “true” frequencies of oscillations of the thread from “formal” ones. We shall explain in detail what we have in mind here, when analyzing an analogous problem for torsional oscillations of shafts. Let us return again to the first problem, when the ends of the thread are secured, and carry through to the end all the necessary calculations for this particular case, in which the masses of the beads are equal and the beads subdivide the thread into equal parts; i.e., m1 = m2 = · · · = mn = m, l0 = l1 = · · · = ln =

l . n+1

In this case we obtain from (18) and (19): m 2 y˙ , 2 i=1 i n

T =

V =

n σ(n + 1)  (yi+1 − yi )2 2l i=0

(y0 = yn+1 = 0).

The Lagrange equations now assume the form m¨ yi −

σ(n + 1) (yi+1 − 2yi + yi−1 ) = 0 (i = 1, 2, . . . , n; y0 = yn+1 = 0). l

Seeking harmonic oscillations of the form yi = ui sin(pt + α)

(i = 1, 2, . . . , n),

we obtain σ(n + 1) (ui+1 − 2ui + ui−1 ) = 0 l (i = 1, 2, . . . , n; u0 = un+1 = 0). σ(n + 1) and putting Dividing the left hand sides of these equations by l −mp2 ui −

2−

ml p2 = 2 cos θ, σ(n + 1)

(22)

(23)

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III. SYSTEMS WITH n DEGREES OF FREEDOM

where θ is a real or complex number, we reduce (22) to the form ui−1 − 2ui cos θ + ui+1 = 0 (i = 1, 2, . . . , n; u0 = un+1 = 0).

(24)

Disregarding first the boundary conditions u0 = un+1 = 0,

(25)

we can represent the general solution of the system (24) in the form ui = C1 cos iθ + C2 sin iθ.

(26)

From the boundary conditions (25) we then find that C1 = 0 and C2 sin(n + 1)θ = 0; hence

πj (j = ±1, ±2, . . . ). n+1 Consequently, by (23), the frequencies p1 < p2 < · · · < pn are obtained from the formula  σ(n + 1) πj sin (j = 1, 2, . . . , n). (27) pj = 2 lm 2(n + 1) From (26) and C1 = 0 it follows that the amplitude vector uj = (u1j , . . . , unj ), corresponding to the frequency pj (j = 1, 2, . . . , n), has coordinates θ=

uij = cj sin

πij n+1

(i = 1, 2, . . . , n).

(28)

Since the oscillation in the general case is written in vector form as y=

n 

uj (Aj cos pj t + Bj sin pj l),

j=1

we obtain, by passing here to coordinates, yi =

n  (Aj cos pj t + Bj sin pj t) sin j=1

πij n+1

(i = 1, 2, . . . , n),

(29)

where pj (j = 1, 2, . . . , n) are the frequencies determined from (27). The constants Aj and Bj (j = 1, 2, . . . , n) are determined from the initial data yi = yi0 ,

y˙ i = y˙i0

for t = 0;

namely, from (29) we obtain: yi0 y˙i0

⎫ πij ⎪ ⎪ ,⎪ = Aj sin ⎪ ⎪ n + 1 ⎬ j=1 n 

πij ⎪ ⎪ ⎪ ⎪ = pj Bj sin ⎪ ⎭ n + 1 j=1 n 

(i = 1, 2, . . . , n).

(30)

It is easy to see for the amplitude vectors uj (j = 1, 2, . . . , n) the following equalities hold  n n  (j = k), πik πij 2 sin = sin n+1 n+1 0 (j = k), i=1

2. OSCILLATIONS OF STURM SYSTEMS

123

which, when j = k, express simply the orthogonality of the vectors uj (j = 1, 2, . . . , n). From this and from (30) it follows that ⎫ n 2 0 πij ⎪ ⎪ Aj = , ⎪ yi sin ⎬ n i=1 n+1 ⎪ (j = 1, 2, . . . , n). (31) n 2  0 πij ⎪ ⎪ ⎪ ⎪ Bj = y˙ sin npj i=1 i n + 1⎭ This solves the problem completely. Following Lagrange, let us now show how, by passing to the limit, to obtain from these results the frequencies of the harmonic oscillations, as well as the general shape of free oscillations of a homogeneous string. Thus, assume we are given a string5 S of length l, the ends of which are fastened. We replace the string by a thread Sn of the same length l with fastened ends, placing on this thread at equal distances of l/(n + 1), beads of mass ρl , n where ρ is the mass per unit length of the string, and n is the number of beads. Passing now to the limit as n → ∞ in the formulas for the thread Sn , we will obtain exact solutions for the string. (n) We begin with the frequencies. According to (27), the frequency pj (j = 1, 2, . . . , n) of the thread Sn is given by  πj 2 σ (n) (n + 1)n sin (j = 1, 2, . . . , n). pj = l ρ 2(n + 1) m=

Passing to the limit as n → ∞, we obtain the well-known formula for the frequencies pj of the string:  σ πj · (j = 1, 2, . . . ). pj = ρ l which shows that the frequencies of the overtones are integral multiples of the frequency of the fundamental tone and the that frequency of each tone is directly proportional to the square root of the tension and inversely proportional to the length of the string and to the square root of the density (Mersenne’s law). We now denote by uj (x) (0 ≤ x ≤ l) the amplitude at the point x of the string S in the j-th harmonic oscillation: y(x, t) = uj (x) sin(pj t + α). The coordinate uij of uj expresses approximately the quantity uj (il/(n + 1)). Consequently we can write uj (x) = lim uij

as n → ∞ if

il →x n+1

(0 ≤ x ≤ l).

Inserting the expression for uij from (28) we obtain uj (x) = cj sin 5A

πjx l

(j = 1, 2, . . . ).

string differs from an ideally flexible thread in that it has a mass.

(32)

124

III. SYSTEMS WITH n DEGREES OF FREEDOM

Thus, the string assumes the shape of a sine wave in harmonic oscillations. Making the same transition to the limit when n → ∞ and

i x → n l

(0 ≤ x ≤ l)

in (29) we obtain an expression for the general oscillation in the form of an infinite series (naturally, such a transition to the limit in a sum with an unlimited number of terms calls for special justification): y(x, t) =

 ∞   πja πja πjx Aj cos t + Bj sin t sin , l l l j=1

 where

a=

σ . ρ

(33)

With this, if a transition to the limit is also made in formulas (31), we obtain ⎫  2 l 0 πjx ⎪ ⎪ dx, ⎪ Aj = y (x) sin ⎬ l 0 l (j = 1, 2, . . . , n), (34)  l ⎪ 2 πjx ⎪ dx⎪ Bj = y˙ 0 (x) sin ⎭ lpj 0 l where y 0 (x) (0 ≤ x ≤ l) is the initial shape of the string, and y˙ 0 (x) (0 ≤ x ≤ l) is the initial velocity of the string at the point with abscissa x. Since it follows from (33) that y(x, 0) = y 0 (x) =

∂ y(x, t) ∂t

j=1

 = y˙ 0 (x) = t=0

∞ 

∞  j=1

Aj sin

πjx , l

Bj pj sin

πjx l

(the second formula obtained by formal differentiation of the series (33)), formulas (34) coincide with Euler’s formulas for Fourier coefficients 2. Torsional oscillations of shafts. Imagine a shaft on which several disks are fastened rotating together with the shaft, and assume that this shaft can rotate uniformly. Consider the motion of the shaft, due to a certain initial impulse. Were the shaft absolutely stiff, the angular velocities and angles of turn of all disks would be the same as the shaft moves. But since this is actually not so, different disks will rotate relative to each other and relative oscillations of discs will be superimposed on the rotation of the shaft as a whole. Let us set up the equation for the kinetic and potential energies of these oscillations. For this purpose we denote by I1 , I2 , . . . , In the moments of inertia of the disks, and by ϕ1 , ϕ2 , . . . , ϕn their angles of rotation. Then, if we neglect the kinetic energy of the shaft itself, assuming its moment of inertia to be very small compared with the moments of inertia of the disks, the kinetic energy of the system is given by the expression: n 1 T = Ij ϕ˙ 2j . (35) 2 j=1

2. OSCILLATIONS OF STURM SYSTEMS

125

Let us now denote by k1 , k2 , . . . , kn−1 the torsional stiffness of the successive portions of the shaft between the disks. Then the potential energy will have the expression: n−1 1 V = kj (ϕj+1 − ϕj )2 . (36) 2 j=1 From the expressions (35) and (36) for T and V we see that we are dealing with a Sturm system. This system is mathematically identical to the system we have obtained for the vibration of a thread with beads, with freely sliding ends, loaded by masses (for exact correspondence it is necessary to replace the indices 0, 1, . . . , n + 1 in (21) by 1, 2, . . . , n, replace the coefficients mj by Ij , and the coefficients lj by kj ). Proceeding to a detailed analysis of this system, we set up the Lagrange equations for it and obtain: ⎫ I1 ϕ¨1 + k1 (ϕ1 − ϕ2 ) = 0,⎪ ⎪ ⎪ I2 ϕ¨2 − k1 (ϕ1 − ϕ2 ) + k2 (ϕ2 − ϕ3 ) = 0,⎬ . (37) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .⎪ ⎪ ⎪ ⎭ In ϕ¨n − kn−1 (ϕn−1 − ϕn ) = 0 Incidentally, these equations could be directly written down by using the law of moments and noticing that the twisting moments −ki−1 (ϕi − ϕi−1 ) and −ki (ϕi − ϕi+1 ) act on the intermediate disk numbered i turning it to the right and to the left respectively; for the outermost disks, one of these moments is dropped. If we now set up the equations for the coordinates of the amplitude vector u of the harmonic oscillation ϕ = u sin(pt + α), (38) they will have the form: I1 p2 u1 − k1 (u1 − u2 ) = 0, I2 p2 u2 + k1 (u1 − u2 ) − k2 (u2 − u3 ) = 0, ....................................... In p2 un + kn−1 (un−1 − un ) = 0. Obviously, these equations are satisfied when p = 0 and u1 = u2 = · · · = un = 0. But the “frequency” p = 0 no longer corresponds to an oscillation (38), but rather to the rotation of the system as a whole: ϕi = ωt + ϕ0

(i = 1, 2, . . . , n).

Of course, there is no contradiction with the statement of Section 1 that λ = p2 > 0, because our system is unstable (the shaft will rotate in one direction by inertia after a small impact; we neglect the effect of friction forces). Since the form V is non-negative and of rank n−1, all the remaining frequencies pi > 0 (i = 2, 3, . . . , n). Thus the general form of the motion of the system of n disks is expressed in vector form as follows: ϕ = (ωt + ϕ0 )e +

n  i=2

ui (Ai cos pi t + Bi sin pi t),

126

III. SYSTEMS WITH n DEGREES OF FREEDOM

where e = (1, 1, . . . , 1); that is, e , u1 = √ I 1 + · · · + In because the normalization conditions for the vector u can now be written in the form: I1 u21 + I2 u22 + · · · + In u2n = 1. Since in the investigation of normal Jacobi form we did not impose any limitations on the signs of the eigenvalues of these forms, Theorem 1 of this section applies to our system. In its application it is necessary, of course, to consider the “frequency” p1 = 0 as the fundamental tone, although the “true” fundamental tone is the frequency p2 , the “true” first overtone is the frequency p3 , etc. Thus, by Theorem 1, the true fundamental tone has one node, the true first overtone has two nodes, etc. (with this, of course, the nodes of two adjacent tones alternate). Proceeding in a somewhat different manner, we could set up equations which would be satisfied only by the “true” oscillations. By adding equations (37), we get I1 ϕ¨1 + I2 ϕ¨2 + · · · + In ϕ¨n = 0, and hence I1 ϕ˙ 1 + I2 ϕ˙ 2 + · · · + In ϕ˙ n = C.

(39)

This equation, incidentally, could be written down at once, since it expresses the fact that the principal angular momentum of free motions of the shaft with the disks, remains constant. Assuming that at the initial instant t = 0 ϕ1 = ϕ2 = · · · = ϕn = ϕ0 ,

and ϕ˙ 1 = ϕ˙ 2 = · · · = ϕ˙ n = ω,

we obtain from (39) (I1 + I2 + · · · + In )ω = C. 0

We can now assume ϕ = 0 and ω = 0, otherwise we would replace ϕi by ϕi − ωt − ϕ0 (i = 1, 2, . . . , n). But then I1 ϕ1 + I2 ϕ2 + · · · + In ϕn = 0. Considering this equation as a constraint imposed on our system, and using it to eliminate one of the angles ψi (i = 1, 2, . . . , n) from the expressions for T and V , we obtain a system with n − 1 degrees of freedom, the frequencies of which coincide with the frequencies of the true oscillations of our system of disks. But this procedure is inconvenient for the investigation of the properties of our system, for the form V loses its Jacobian character, and the form T will contain products of angles, if one angle is eliminated. Let us finally point out that if one of the cross sections of the shaft is secured so that it remains stationary (or, in the more general case, it is in a state of uniform rotation, in which case the angle of rotation of the disks should be measured from the angle of rotation of this cross section), there will be no zero frequencies in the system, since it will be stable. Assume, for example, that the left end of the shaft is rigidly fastened (we assume that the disks are numbered from left to right). In this case, instead of

2. OSCILLATIONS OF STURM SYSTEMS

127

expression (36), we obtain the expression 1 1 k0 ϕ21 + ki (ϕi+1 − ϕi )2 , 2 2 i=1 n

V =

where k0 is the stiffness of the section of the shaft between the left end and the first disk. Naturally, here V is a positive form and consequently all frequencies are positive. Thus, in this case6 the fundamental tone will not have any nodes. That is, all disks will simultaneously rotate in one direction; the first overtone will have exactly one node, so that the disks will break up into two successive groups, one of which will rotate in one direction and the other in the opposite direction, etc. 3. Small oscillations of a compound mathematical pendulum. Let P1 Pn+1 be an ideally flexible massless thread which is freely suspended at the point Pn+1 . Let, at the points P1 , P2 , . . . , Pn of this thread, be placed masses m1 , m2 , . . . , mn . We investigate small oscillations of this system of masses (compound pendulum) in the plane Y Z, where the Z-axis passes through the vertical position of equilibrium of the thread; here we can neglect vertical displacements of the masses and consider only transverse displacement, parallel to the horizontal axis Y . We denote by li the length of the section of the thread Pi Pi+1 , and by σi its tension (i = 1, 2, . . . , n); it is obvious that σi = g

i 

mk

(i = 1, 2, . . . , n),

k=1

where g is the acceleration of gravity. The kinetic energy has the expression: 1 mi y˙i2 , 2 i=1 n

T =

and the potential energy, on the basis of the same considerations as those given on p. 120), can be written as 1  σi (yi+1 − yi )2 2 i=1 li n

V =

(yn+1 = 0).

We see that our problem differs very little from the problem of determining the free transverse oscillations of a thread with beads, and, naturally, again brings us to a Sturm system. For the coordinates ui (i = 1, 2, . . . , n) of the amplitude vector of the harmonic oscillation y = u sin(pt + α), we now have the following equation:   σi−1 σi σi−1 σi 2 − ui − ui−1 + + − mi p ui+1 = 0. (40) li−1 li li−1 li (i = 1, 2, . . . , n; σ0 = 0; u0 = un+1 = 0). 6 Incidentally, this case is mathematically identical to that of a vibrating string with beads, one end of which can slide freely in a direction perpendicular to the thread, and the other rigidly fastened.

128

III. SYSTEMS WITH n DEGREES OF FREEDOM

Let us consider the special case of equally spaced identical masses. In this case σi = mgi (i = 1, 2, . . . , n), and equations (40) assume the form (i − 1)ui−1 + (ξ − 2i + 1)ui + iui+1 = 0 (i = 1, 2, . . . , n; u0 = un+1 = 0),

(41)

where ξ = p2 l/ng. On the other hand, the Chebyshev–Posse polynomials7 : Lk (ξ) =

1 ξ dk e (ξ k e−ξ ) = 1 − k! dξ k

  2   ξk k ξ k ξ + − · · · + (−1)k 2 2! 1 1! k!

(k = 0, 1, . . . ) satisfy the following relations: kLk−1 (ξ) + (ξ − 2k − 1)Lk (ξ) + (k + 1)Lk+1 (ξ) = 0

(42)

(k = 0, 1, 2, . . . ). Comparing (41) and (42), we find that  ui = const × Li−1 (ξ) = const × Li−1

p2 l ng

 (i = 1, 2, . . . , n + 1).

As un+1 = 0, we obtain the following equation for the frequencies pj (j = 1, . . . , n):  Ln

p2 l ng

 = 0.

Therefore the general form of the oscillations of our pendulum can be written as yi =

n  j=1

 cj Li−1

p2j l ng

sin(pj t + αj )

(i = 1, 2, . . . , n).

Earlier we obtained a solution of the problem of free oscillations of a homogeneous string as a limit of the solution of the problem of oscillations of a thread with equidistant beads. Similarly, the problem of oscillation of n-fold composite pendulum with equal and equidistant masses permits us, by passing to the limit, to solve the problem of small oscillations of a heavy homogeneous chain, which is suspended at one end. To do this one has to use the connection which exists between the Chebyshev–Posse polynomials and Bessel functions of order zero. We refer to the article of O. Bottema [6] for details. Let us also notice that using other choices of the masses mi and the distances li (i = 1, 2, . . . , n) one can achieve that the solution of the problem of oscillations of a composite pendulum be expressed in terms of other classical orthogonal polynomials, for example the Chebyshev–Hermite polynomials. 7 These polynomials have until recently been incorrectly called Laguerre polynomials (concerning this subject, see the interesting survey by Ya. L. Geronimus [18]).

SECOND METHOD OF SETTING UP THE EQUATIONS OF SMALL OSCILLATIONS 129

3. Second method of setting up the equations of small oscillations of mechanical systems Consider transverse oscillations of a system of masses on a beam. We neglect the mass of the beam, and assume that there are n concentrated masses m1 , m2 , . . . , mn placed at sections (1), (2), . . . , (n) of the beam. We want to find the free oscillations of this system resulting from the bending of the beam. Let us denote by y1 , y2 , . . . , yn the deflections at the sections (1), (2), . . . , (n), counted from the equilibrium position of the system. Then the kinetic energy will be n 1 T = mi y˙i2 2 i=1 To find the potential energy V =

n 1  aik yi yk , 2

(43)

i,k=1

one has to find that energy which is stored in the beam bent under the action of the forces F1 , . . . , Fn which are concentrated at the sections (1), (2), . . . , (n), and causing in these sections the deflections y1 , . . . , yn . For this purpose one has to use the bending equation of the beam and the conditions of fastening at the ends. In the general case, the expression for the potential energy is not simple to calculate; we only note that always aik = 0

for |i − k| > 2.

As concerns the coefficients aik for |i−k| ≤ 2, these are, generally speaking, different from zero. Thus, this mechanical system is not a Sturm system. At the same time, it has, as will be shown below, all the basic properties of Sturm systems and in particular, Theorem 1 is applicable to it. A complete investigation of the problem under consideration will be carried out in a different place (see Sections 6 and 8). We shall merely indicate here that our procedure will be to investigate not the matrices A = aik n1 , but the inverse matrices A−1 = αik n1 . The Lagrange equations for our system have the form mi y¨i +

n 

aik yk = 0

(i = 1, 2, . . . , n).

(44)

k=1

Solving this system with respect to yi (i = 1, 2, . . . , n), we get yi = −

n 

αik mk y¨k

(i = 1, 2, . . . , n),

(45)

k=1

where αik n1 is the inverse matrix of the matrix aik n1 . Thus, for harmonic oscillations yi = ui sin(pt + α) (i = 1, 2, . . . , n) we obtain the equations n  k=1

αik mk uk =

1 ui p2

(i = 1, 2, . . . , n),

130

III. SYSTEMS WITH n DEGREES OF FREEDOM

and, consequently, the squares of the frequencies p2 are reciprocals to the eigenvalues of the matrix αik mk n1 . In the next section we shall show that the matrix αik n1 , and thus the matrix αik mk n1 , is oscillatory. From this we shall derive the oscillatory properties of the vibrations of our system. The matrix αik n1 has a simple mechanical meaning. If the forces F1 , . . . , Fn concentrated in the sections (1), . . . , (n) cause deflections y1 , . . . , yn , then by the linearity of the relations between yk (k = 1, 2, . . . , n) and Fi (i = 1, 2, . . . , n), the potential energy V , stored in the beam, is expressed by 1 Fi yi 2 i=1 n

V =

(46)

(more about this in the next section). Comparing this expression with (43) we see8 that n  aik yk (i = 1, 2, . . . , n). (46 ) Fi = k=1

Solving this system with respect to yi (i = 1, 2, . . . , n) we find that yi =

n 

αik Fk

(i = 1, 2, . . . , n).

(46 )

k=1

If we set all Fk = 0, with exception of a certain Fj , which we put equal to 1, we find that yi = αij (i = 1, 2, . . . , n). Thus, the coefficient αij (i, j = 1, 2, . . . , n) is the deflection at the section (i) caused by the action of a concentrated unit force applied to the section (j). Coefficients αij are called the influence coefficients of the beam. Assume that the central axis of the beam is located along the x axis, for example, from the point 0 to the point x = l, and let us introduce the influence function K(x, s) (0 ≤ x, s ≤ l) of the beam, where K(x, s) is the deflection of the cross-section having the abscissa x under the influence of a single concentrated unit force applied to the section with abscissa s. Then αij = K(si , sj )

(i, j = 1, 2, . . . , n),

where si is the abscissa of the section (i) (i = 1, 2, . . . , n). We see that to study the structure of the matrix of influence coefficients αij n1 , it is necessary first to study the structure of the influence function K(x, s), which we do in the next section. Let us note that the equations of motion (45) offer still another convenience compared with equations (44) in that they generalize naturally to integral equations in cases when the masses are distributed continuously over the beam (see Chapter IV). Let us call attention to still another circumstance. Using equations (46), (46 ) and (46 ), we can write expressions for the potential energy in terms of the deflections and in terms of the concentrated forces in the form of two positive quadratic 8 This argument is not quite rigorous; we shall give a more rigorous argument in the next section.

4. INFLUENCE FUNCTIONS

131

forms: V =

V =

n 1 1  aik yi yk = A(y, y), 2 2

1 2

i,k=1 n  i,k=1

αik Fi Fk =

1 −1 A (F, F ). 2

In comparing potential energies of the two systems S and S1 , it is always necessary to stipulate whether the energies are compared for equal deflections or for equal forces since, as was clarified in Theorem 12 of Chapter I, Section 9, the inequality A(y, y) ≥ A1 (y, y) implies the inequality A−1 (F, F ) ≤ A−1 1 (F, F ). Thus, for example, of two different fastenings of the ends of the beam, that one which has greater potential energy under the same strains; i.e., under the same deflections, is called the stiffer one. If we use in the comparison the same forces, however, then the stiffer system will have lower potential energy. It is precisely this force characteristic that is convenient to use as a definition of the more rigid system in the case of systems with infinite number of degrees of freedom, for in this case it is not always possible to compare the potential energies under the same strains: possible strains under certain fastenings may be impossible under others. 4. Influence functions 1. Let S denote an elastic linear continuum, for example a string, a rod, a multiple-span beam, a lattice made of rods or strings, etc. We will assume that the points of this continuum are capable of displacement parallel to certain fixed direction under the influence of forces parallel to the same direction. For convenience in the further exposition, we shall start with a scheme of transverse displacements parallel to the Y axis, and in accordance with this we shall denote the displacement of the point x by y(x) and call it the deflection. However all our arguments retain their force for the general case (in particular, for the case of longitudinal displacements). In this section we shall denote by x, s, etc. the points of the continuum itself (and not their coordinates). In the case of a continuum which extends along an axis, we shall simultaneously mean by x, s, . . . the coordinates of the corresponding points. A point of a continuum, regardless of whether there is or there is no constraint at this point, we shall call movable if it is capable of displacement. Thus, for example, a point at which an elastic support is placed will be called movable. We denote by K(x, s) the deflection of the point x caused by the action of a unit force applied to the point s. The function K(x, s) is called the influence function (of the given elastic continuum). We shall assume that all our displacements are sufficiently small; consequently, the law of independence of the action of forces applies. Thus, the concentrated forces F1 , F2 , . . . , Fn applied at the points s1 , s2 , . . . , sn will produce at the point x

132

III. SYSTEMS WITH n DEGREES OF FREEDOM

the deflection y(x) =

n 

Fj K(x, sj ).

(47)

j=1

We assume further that the continuum S is a conservative system. Then the work V necessary to put S to the state (47) is completely determined by the shape y(x) and consequently will depend only on the quantities Fj (j = 1, 2, . . . , n). Let V = V (F1 , . . . , Fn ) be the potential energy which is stored in the continuum S in the state (47). We now subject the forces Fj (j = 1, 2, . . . , n) to elementary increments dF1 , . . . , dFn ; then the deflections yi = y(si ) acquire increments dyi : dyi =

n 

K(si , sj )dFj

(i = 1, 2, . . . , n).

j=1

Consequently, for the corresponding increment dV , which is equal to the elementary work of the forces F1 , . . . , Fn on the displacements dy1 , . . . , dyn , we obtain the expression: dV =

=

n 

Fi dyi =

i=1

 n

n 

K(si , sj )Fi dFj =

i,j=1



K(si , s1 )Fi dF1 +

 n

i=1

hence



(48)

K(si , s2 )Fi dF2 + . . . ;

i=1

 ∂V = K(si , sj )Fi ∂Fj i=1 n

(j = 1, 2, . . . , n).

Writing further ∂2V ∂2V = ∂Fi ∂Fj ∂Fj ∂Fi

(i, j = 1, 2, . . . , n),

we obtain K(si , sj ) = K(sj , si )

(i, j = 1, 2, . . . , n).

(49)

Since si (i = 1, 2, . . . , n) are arbitrary points of the continuum S, we deduce that the function K(x, s) is symmetric; i.e., K(x, s) ≡ K(s, x).

(50)

This equation expresses the so-called Maxwell reciprocity principle: The deflection at the point x under the influence of a unit force applied at the point s is equal to the deflection at the point s under the influence of a unit force applied at the point x. Equations (49) are nothing but the condition that the right hand side of (48) be an exact differential. Integrating (48) subject to the condition V (0, 0, . . . , 0) = 0, we obtain: n 1  V = K(si , sj )Fi Fj . (51) 2 i,j=1

4. INFLUENCE FUNCTIONS

133

We now assume that all the points s1 , . . . , sn are movable. Since the energy V is by its very meaning a positive quantity (if not all Fi = 0 (i = 1, 2, . . . , n)) we reach the conclusion that the quadratic form n 

K(si , sj )Fi Fj

(52)

i,j=1

is positive, no matter how the movable points s1 , . . . , sn are chosen. Let us introduce the following notation:      K(x1 , s1 ) . . . K(x1 , sn )  x1 x2 . . . xn  K(x2 , s1 ) . . . K(x2 , sn )  K =  s1 s2 . . . sn  ..........................    K(xn , s1 ) . . . K(xn , sn ) (n is an arbitrary natural integer; x1 , . . . , xn , s1 , . . . , sn are arbitrary points of the continuum). As is known from Section 7 of Chap. I, necessary and sufficient conditions for the positiveness of the form (52) can be expressed by the following inequalities:     s1 s2 . . . sn s 1 s2 K(s1 , s1 ) > 0, K > 0, . . . , K > 0. s1 s2 s1 s2 . . . sn Since here the movable points s1 , s2 , . . . , sn of the continuum and their number n are arbitrary, we arrive at the inequality   x 1 x2 . . . xn K > 0, (53) x1 x2 . . . xn which holds for any movable points x1 , x2 , . . . , xn (n = 1, 2, . . . ). We notice in particular the movability of the point x = c is expressed by the inequality K(c, c) > 0. 2. Let us show now that inequalities (53) have by themselves a simple mechanical meaning. We introduce hinged supports at p arbitrary movable points c1 , c2 , . . . , cp . We denote by S ∗ the continuum with these new additional constraints, and by K ∗ (x, s) the corresponding influence function (i.e., the deflection at the point x of the continuum S ∗ under the influence of a unit force concentrated at the point s). Let us derive an expression for K ∗ (x, s). For this purpose we replace the supports at the points c1 , c2 , . . . , cp by their reactions R1 , . . . , Rp . Then K ∗ (x, s) will give us the deflection of the point under the influence of the forces R1 , . . . , Rp and unit force applied at the point s. Hence K ∗ (x, s) = K(x, s) +

p 

Rj K(x, cj ).

(54)

i=1

The reactions Rj (j = 1, 2, . . . , p) are determined from the conditions that the deflection of S ∗ at the points ci (i = 1, 2, . . . , p) is equal to 0; i.e., from the conditions p  K ∗ (ci , s) = K(ci , s) + Rj K(ci , cj ) = 0 (i = 1, 2, . . . , p). (55) j=1

134

III. SYSTEMS WITH n DEGREES OF FREEDOM

To eliminate the reactions Rj from (54) and (55), we present these equations in the following form (putting R0 = 1): R0 [K(x, s) − K ∗ (x, s)] + R0 K(c1 , s) +

p 

Rj K(x, cj ) = 0,

j=1 p 

Rj K(c1 , cj ) = 0,

j=1

........................................... p  R0 K(cp , s) + Rj K(cp , cj ) = 0. j=1

Equating the determinant of this homogeneous system to 0, we find    K(x, s) − K ∗ (x, s) K(x, c1 ) . . . K(x, cp )    K(c1 , s) K(c1 , c1 ) . . . K(c1 , cp )     = 0,  .............................................    K(cp , s) K(cp , c1 ) . . . K(cp , cp ) hence

 x c1 . . . cp s c1 . . . cp  .  c 1 . . . cp K c1 . . . cp 

K K ∗ (x, s) =

(56)

We denote by γn (n = 1, 2, . . . , p) the deflection produced at the point cn if a unit force is applied at this point and if pivoted supports are introduced at the point c1 , c2 , . . . , cn−1 . From (56), after replacing p by n − 1, we get   c 1 . . . cn K c1 . . . cn . γn =  c1 . . . cn−1 K c1 . . . cn−1 Consequently, by (53), we have γn > 0. Thus, inequalities (53) imply the following mechanically obvious fact: independent of how many hinged supports are introduced, when a single concentrated force is applied to S, the deflection at the point of application of the force (if this point is movable) differs from 0 and is directed along the action of the force. Conversely, this simple mechanical fact results in inequality (53). Indeed, it follows from it first of all that K(c1 , c1 ) = γ1 > 0, and then

 K

c1 c1

c2 c2

 = γ1 γ2 > 0,

and so on. It is also mechanically obvious that γn (the deflection produced at the point cn when a unit force is applied there, after the supports are introduced at the points

4. INFLUENCE FUNCTIONS

135

cn , c2 , . . . , cn−1 ) will be less than the deflection K(cn , cn ) at the point cn prior to the introduction of the supports, and consequently 

c1 c1

c2 c2

... ...

cn cn

From this follows  c1 K c1

c2 c2

... ...

cn cn

K



 < K(cn , cn )K

... ...

c1 c1

cn−1 cn−1

 .

 < K(c1 , c1 )K(c2 , c2 ) . . . K(cn , cn ).

(57)

With the aid of analogous arguments, introducing successively two groups of supports c1 , . . . , cn ; d1 , . . . , dn , one can obtain the inequality  K

c1 c1

... ...

cm cm

d1 d1

... ...

dn dn



 0 (62) ϕ(x) = i=1

i=1

vanishes on the interval I at most n − 1 times. We will need the following characterization of Chebyshev systems (see the monograph by S.N. Bernshtein [4]; many other properties of these systems can be found there too.) Lemma 2. In order for continuous functions ϕ1 (x), ϕ2 (x), . . . , ϕn (x) (x ∈ I) to constitute a Chebyshev system on I, it is necessary and sufficient that the determinant   ϕ 1 ϕ2 . . . ϕn Δ x1 x2 . . . xn be different from 0 (and consequently, of constant sign) for all x1 < x2 < · · · < xn from I. Proof. The determinant vanishes at certain points x1 < x2 < · · · < xn from I if and only if the system of equations n 

ci ϕi (xk ) = 0 (k = 1, 2, . . . , n)

i=1

has a non-trivial solution (c1 , c2 , . . . , cn ); the linear combination (62) corresponding to this solution will vanish at the points x1 , x2 , . . . , xn . The condition that the determinant Δ is not zero on the set M of n-dimensional points x1 < x2 < · · · < xn from I implies the constancy of sign of Δ on M , since M is a connected set and Δ is a continuous function on M . This proves the lemma. Let there be given a function ϕ(x), defined in an interval (a, b), and vanishing at an interior point c in this interval. The point c will be called a node of the function ϕ(x), if in every neighborhood of c, on opposite sides of c, there exist two points x1 and x2 (x1 < c < x2 ) such that ϕ(x1 )ϕ(x2 ) < 0. We shall call the point c an antinode of the function ϕ(x) if in every sufficiently small two-sided neighborhood of c the function ϕ(x) does not change sign, and on each side of c, it is not identically equal to 0. Thus, each isolated zero12 of the function in (a, b) is either a node of an antinode. From Definition 2 and Lemma 2 follows: 12 i.e.,

such a zero that there are no other zeros in a neighborhood of it.

138

III. SYSTEMS WITH n DEGREES OF FREEDOM

Corollary. If functions ϕ1 (x), ϕ2 (x), . . . , ϕn (x) form a Chebyshev system in an interval I, and a function   n n  ϕ(x) = ci ϕi (x) c2i > 0 i=1

i=1

has r zeros in I, including q antinodes, then r+q ≤n−1 (i.e., in the estimate of the number of zeros each antinode can be counted as two zeros). Proof. 13 We shall agree to say that s points x1 < x2 < · · · < xs from I have the property Z, if for some integer h (−1)h+k ϕ(xk ) ≥ 0 (k = 1, 2, . . . , s), holds; i.e., the maximum number of sign changes in the sequence ϕ(x1 ), . . . , ϕ(x3 ) is s − 1. If points x1 , x2 , . . . , xs have the property Z and α is an antinode of ϕ(x) which is different from these s points, then it is possible to complement this system of points with two points α − ε and α, or α and α + ε (where ε is a sufficiently small positive number), so that the resultant s + 2 points again have property Z. Indeed, let xg < α − ε < α < α + ε < xg+1 ,

ϕ(α − ε)ϕ(α + ε) > 0,

If (−1)h+g+1 ϕ(α − ε) > 0, then (−1)h+g ϕ(xg ) ≥ 0,

(−1)h+g+1 ϕ(α − ε) > 0, (−1)h+g+3 ϕ(xg+1 ) ≥ 0;

(−1)h+g+2 ϕ(α) = 0, in the opposite case, (−1)h+g ϕ(xg ) ≥ 0,

(−1)h+g+1 ϕ(α) = 0,

(−1)h+g+2 ϕ(α + ε) > 0,

(−1)h+g+3 ϕ(xg+1 ) ≥ 0,

and similar inequalities will hold if α < x1 or xs < α. Let us show that there always exist r + q + 1 points x1 < x2 < · · · < xr+q+1 in I having the property Z if r is the number of zeros of the function ϕ on the interval I and q is the number of antinodes. Indeed, let the function ϕ(x) have p nodes in (a, b). Then there exist points β1 < β2 . . . , < βp+1 inside (a, b) with ϕ(βk )ϕ(βk+1 ) < 0 (k = 1, 2, . . . , p). Since the points β1 , β2 , . . . , βp+1 have property Z and ϕ(x) has q antinodes in (a, b), it is possible to find a system of p+2q +1 points within (a, b) having the property Z. We shall complement this system with zeros of the function ϕ(x), at the endpoints of the interval I, if there are such zeros, and in this way obtain the desired system of r + q + 1 points x1 < x2 < · · · < xr+q+1 from I. Let us assume that r + q ≥ n. Then for an integer h we have (−1)h+k ϕ(xk ) ≥ 0 (k = 1, 2, . . . , n + 1). 13 For

another proof, see [4].

(63)

5. CHEBYSHEV SYSTEMS OF FUNCTIONS

139

Since the function ϕ(x) is a linear combination of the functions ϕ1 (x), ϕ2 (x), . . . , ϕn (x),  . . . ϕn ϕ = 0. . . . xn xn+1 elements of the last row, which contains the



then

ϕ 1 ϕ2 Δ x1 x 2 Expanding this determinant by the values of the function ϕ(x), we get  n+1  ϕ1 (−1)n+k+1 ϕ(xk )Δ x1

ϕ2 . . . . . . . . . . . . . . . . . . . . . . xk−1 xk+1 . . .

k=1

ϕn xn+1

 = 0.

According to (63) and Lemma 2, the sum in the left hand side of this equation has all terms of the same sign. Therefore all the terms vanish. Furthermore, according to this lemma, all the determinants Δ are different from 0, so all values of the function ϕ(x) at the points x1 , x2 , . . . , xn+1 should vanish. This is impossible since the functions ϕ1 (x), ϕ2 (x), . . . , ϕn (x) form a Chebyshev system and therefore, the number of zeros of the function ϕ(x) does not exceed n − 1. We have therefore proved that always r + q ≤ n − 1. 3. Lemma 3. Let a continuous function ϕ(x), not identically equal to 0 on the interval [a, b], change sign in this interval at most n − 1 times, and let L(x, s) (a ≤ x, s ≤ b) be a continuous kernel, having the property     x 1 x2 . . . xn x1 < x2 < · · · < xn L >0 a≤ ≤b . s1 s2 . . . sn s1 < s2 < · · · < sn Then the function



b

L(x, s)ϕ(s)ds

Φ(x) = a

vanishes on the interval [a, b] at most n − 1 times. Here we say that the function ϕ(x), defined on the interval [a, b], changes sign k times (which will be written as Sϕ = k), if on the interval [a, b] there are k + 1 points x1 < x2 < · · · < xk+1 such that ϕ(xi )ϕ(xi+1 ) < 0 (i = 1, 2, . . . , k) and there do not exist k + 2 points having this property. If a continuous function on the interval [a, b] changes sign k times, then it has k nodes inside this interval. Proof of Lemma 3. By assumption, there exist points a = ξ0 < ξ1 < ξ2 < · · · < ξn−1 < ξn = b such that in each of the intervals (ξi−1 , ξi )

(i = 1, 2, . . . , n)

the function ϕ(x) does not change sign and is not equal identically to 0. Let us put  ξi Φi (x) = L(x, s)ϕ(s)ds (i = 1, 2, . . . , n) ξi−1

140

III. SYSTEMS WITH n DEGREES OF FREEDOM

Then Φ(x) =

n 

Φi (x).

i=1

For all (a ≤)x1 < x2 < · · · < xn (≤ b) the determinant   ξn    ξ1  x 1 x2 . . . xn Φ 1 . . . Φn = ϕ(s1 ) . . . ϕ(sn )ds . . . dsn ··· L Δ x1 . . . xn s1 s2 . . . sn ξn−1 ξ0 differs from 0, since the integrand is not equal identically to 0 and its non-zero values have the same sign. Thus, the functions Φi (x) (i = 1, 2, . . . , n) form a Chebyshev system in the closed interval [a, b] and consequently the function Φ(x) vanishes in this interval at most n − 1 times. This proves the lemma. We shall use this lemma as applied to the heat kernel: 2 1 Lt (x, s) = √ e−(x−s) /t πt

(t > 0),

which, according to Section 3 of Chap. II, satisfies the condition of the lemma for all n (the heat kernel is totally positive). This kernel has one remarkable property. Let ϕ(x) (a ≤ x ≤ b) be a continuous function. We put  b Φ(x, t) = Lt (x, s)ϕ(s)ds (a ≤ x ≤ b; t > 0). a

If we define the function ϕ(x) as equal to 0 outside the interval [a, b], then we have  +∞  b √ 2 2 1 1 e−(x−s) /t ϕ(s)ds = √ e−ξ ϕ(x + tξ)dξ. Φ(x, t) = √ π −∞ πt a This, as is well-known, implies Φ(x, t) → ϕ(x) as t → +0

(64)

for any interior point x of the interval (a, b). Using this, we shall prove the following lemma which is fundamental for what is to follow. Lemma 4. Let ϕ1 (x), ϕ2 (x), . . . , ϕn (x) be linearly independent continuous functions on the interval [a, b]. In order for every linear combination ϕ(x) =

n 

ci ϕi (x)

i=1

to change sign on the interval [a, b] at most n − 1 times for all choices of coefficients ci (i = 1, 2, . . . , n;

n  i=1

c2i > 0),

5. CHEBYSHEV SYSTEMS OF FUNCTIONS

141

it is necessary and sufficient that the determinant   ϕ1 ϕ2 . . . ϕn Δ x1 x2 . . . xn has the same sign for all possible (a ≤)x1 < x2 < · · · < xn (≤ b), for which it is different from 0. n

Proof. If for some ci (i = 1, 2, . . . , n;

n  i=1

c2i > 0), the function ϕ(x) =

i=1 ci ϕi (x) changes sign on the interval [a, b] at most n − 1 times, then according to Lemma 3, for all t > 0, the function

Φ(x, t) =

n 

ci Φi (x, t)

i=1

has at most n − 1 zeros on the interval [a, b]. In view of (64), it is evident that the converse also holds; that is the indicated property of the function Φ(x, t) for all t > 0 implies the indicated property of the function ϕ(x). It follows that the functions ϕ1 (x), ϕ2 (x), . . . , ϕn (x) will have the property mentioned in Lemma 4 if and only if for every t > 0 the functions Φ1 (x, t), Φ2 (x, t), . . . , Φn (x, t) (a ≤ x ≤ b) form a Chebyshev system in [a, b]. The latter will take place if the determinant   Φ1 Φ 2 . . . Φn for (a ≤)x1 < x2 < · · · < xn (≤ b) Δ x1 x2 . . . xn does not change sign. On the other hand, according to (64)    ϕ1 ϕ 2 . . . ϕn Φ1 = lim Δ Δ x1 x2 . . . xn x1 t→+0

Φ2 x2

... ...

Φn xn



for all (a ≤)x1 < x2 < · · · < xn (≤ b). From this follows that the condition of the lemma is necessary. The sufficiency of this condition follows from the easily proved identity:        b  1 b Φ 1 . . . Φn x 1 . . . xn ϕ1 . . . ϕn Δ = Δ ds1 . . . dsn . ··· Lt x1 . . . xn s1 . . . sn s1 . . . sn n! a a Indeed, if the condition of the lemma is satisfied for (a ≤)x1 < x2 < · · · < xn (≤ b), the integrand does not change sign and does not vanish identically, according to Lemma 1. Consequently, the functions Φ1 (x, t), Φ2 (x, t), . . . , Φn (x, t) for any t > 0 will form a Chebyshev system in [a, b]. It follows from this that the function ϕ(x) = lim Φ(x, t) cannot change sign in [a, b] more than n − 1 times14 . t→+0

14 Lemma

4 can be proved purely algebraically on the basis of Theorem 4 of Sec. 2, Chap. V.

142

III. SYSTEMS WITH n DEGREES OF FREEDOM

6. The oscillatory character of the influence function of a segmental continuum In what follows, I denotes the set of all the movable points of a continuum S 1. In this subsection we shall assume that a linear elastic continuum S is a segmental continuum; i.e., that S has a form of a rectilinear segment and that the interior points S are free of external constraints (intermediate supports, elastic base, etc.). We shall assume that the segmental continuum extends along the X axis from the point x = 0 to the point x = l. Thus, the interval I for a segmental continuum S can be one of the following: 1) a closed interval [0, l] if both ends of S are movable, 2) a semi-closed interval [0, l) or (0, l] if one of the ends 0 or l is movable, and finally, 3) an open interval (0, l) if both ends are non-movable. It is intuitively clear that a segmental continuum has the following two properties, the mathematical justification of which will be deferred to the succeeding sections. Property A. Under the action of one concentrated force, applied to an internal point of a segmental continuum, the deflection of all movable points of the continuum differs from zero and has the same direction (sign) as the force. If we use the influence function K(x, s) (0 ≤ x, s ≤ l) then it is evident that property A can be written as the inequality K(x, s) > 0 (0 < x < l, s ∈ I).

(65)

Naturally, this inequality is satisfied also in that case when x and s coincide with one and the same movable end (cf. (53)). However, this inequality might not hold at x = 0 and s = l, when both of these ends are movable. Thus, for example, if S is a rod, hinged on two elastic supports, and we apply at one of these supports a concentrated force, then the deflection at the other support will be equal to zero; i.e., K(0, l) = 0. Property B. Under the influence of n concentrated forces, the deflection y(x) can change sign at most n − 1 times. Under the action of n concentrated forces F1 , F2 , . . . , Fn , applied at the points (0 ≤)s1 < s2 < · · · < sn (≤ l), the deflection y(x) is given by the expression y(x) =

n 

Fi ϕi (x),

i=1

where ϕi (x) = K(x, si ) (i = 1, 2, . . . , n). In this case

 Δ

ϕ1 x1

ϕ2 x2

... ...

ϕn xn



 =K

x1 s1

x2 s2

... ...

xn sn

 .

(66)

Applying Lemma 4 of the preceding section, we conclude that property B is equivalent to the following property of the influence function: All the determinants (66) with fixed (0 ≤)s1 < s2 < · · · < sn (≤ l) and variable (0 ≤)x1 < x2 < · · · < xn (≤ l) do not change sign.

6. THE OSCILLATORY CHARACTER OF THE INFLUENCE FUNCTION

143

 But when xi = si (i = 1, 2, . . . , n), by the positivity of the quadratic form K(si , sj )Fi Fj , the determinants (66) are positive. Thus all the determinants (66) are non-negative;     x1 x 2 . . . x n x1 < x2 < · · · < xn K ≥0 0≤ ≤l . (67) s1 s2 . . . sn s1 < s2 < · · · < sn Now we introduce the following: Definition 3. An influence function K(x, s) (0 ≤ x, s ≤ l) of a continuum S will be called oscillatory, if the following three conditions are satisfied: 1◦ K(x, s) > 0  x 1 x2 ◦ 2 K s1 s2  x 1 x2 3◦ K x1 x2

... ...

xn sn

... ...

xn xn



for ≥0



0 < x < l; s ∈ I, for

0
0 for x1 < x2 < · · · < xn

in

I.

Using this definition, we can state the following conclusion from the preceding arguments. Theorem 2. An influence function K(x, s) (0 ≤ x, s ≤ l) of a segmental elastic continuum S which has properties A and B is oscillatory. 2. We note that when we derived the oscillatory character of the influence function from the properties A and B, we used the fact that the form (51) which expresses the potential energy which is stored by the continuum S under the static action of concentrated forces is positive. This corresponds to the stability of the equilibrium position of the continuum under the considered conditions of its fastening. In fact, the inequalities (53) were obtained from this. Still another circumstance is of great interest. If it is assumed that the deflection y(x) satisfies a linear differential equation L[y] = f (x)

(0 ≤ x ≤ l),

in which the right hand side f (x) is the intensity of the load at the point x, then property A is obtained as a consequence of property B and from the fact that the energy V is positive. Indeed, let us assume the opposite; i.e., that in the presence of property B there exist two movable points x0 and s0 which do not coincide simultaneously with the ends 0 and l, such that K(x0 , s0 ) = 0. By (53), x0 = s0 ; on the basis of the symmetry of the function K(x, s) we can assume that x0 < s0 . Let us consider first the case when x0 = 0. By property B, the inequalities (67) are satisfied, in particular,  K

x s

x0 s0



K(x, s) ≥ 0 (0 ≤ x, s ≤ l),      K(x, s) K(x, s0 )  x < x0   ≥0 0≤ = ≤l . K(x0 , s) K(x0 , s0 )  s < s0

Hence, since K(x0 , s0 ) vanishes, we find −K(x0 , s)K(x, s0 ) ≥ 0

144

and consequently,

III. SYSTEMS WITH n DEGREES OF FREEDOM

  x < x0 0≤ ≤l . K(x0 , s)K(x, s0 ) = 0 s < s0

(68)

Since x0 < s0 , we can put here s = x0 . But, by the inequalities (53), K(x0 , x0 ) > 0 and therefore it follows from (68) that K(x, s0 ) = 0 for 0 < x < x0 (< s0 ). On the other hand, the function K(x, s0 ), as the deflection due to a concentrated force applied at the point s0 , satisfies on the interval (0, s0 ) a homogeneous differential equation L[y] = 0. But if a function satisfies a linear homogeneous differential equation in an interval and vanishes in a subinterval of this interval, then this function is identically equal to 0 (since the zero function is the unique solution corresponding to zero initial conditions). Therefore K(x, s0 ) = 0 (0 < x < s0 ) and, by the continuity of the deflection K(x, s0 ), K(s0 , s0 ) = 0, which contradicts inequalities (53). If x0 = 0, then s0 = l. Therefore, putting x0 < x ≤ l and s0 < s ≤ l, we obtain a relation analogous to (68):   x0 < x K(x, s0 )K(x0 , s) = 0 ≤l . s0 < s Replacing here x by s0 we find K(x0 , s) = 0 (s0 < s ≤ l). Since K(x, s) (x0 < s < l) is again a solution of the equation L[y] = 0 with zero initial conditions (at the point s = s0 ), then K(x0 , s) = 0 (x0 < s ≤ l) and from this we obtain by continuity K(x0 , x0 ) = 0, which contradicts inequalities (53). This proves our statement. Since the deflection will henceforth always satisfy a linear differential equation, we see that to prove the oscillatory character of an influence function, it is enough to verify property B of the segmental continuum under consideration. 3. Let us notice that the oscillatory character of the influence function K(x, s) makes it possible to prove property B of an elastic continuum in the following stronger form: If forces F1 , F2 , . . . , Fn are applied to a segmental elastic continuum S at successive points (0 0, . . . , Fp > 0; Fp+1 < 0, . . . , Fq < 0; Fq+1 > 0, . . . Fr > 0; Fr+1 < 0, . . . , Fn < 0. Let us form according to these groups, the following functions: ϕ1 (x) =

ϕ2 (x) = ϕ3 (x) = ϕ4 (x) =

p 

j=1 q  j=p+1 r  j=q+1 n 

K(x, sj )Fj ,

K(x, sj )Fj , K(x, sj )Fj , K(x, sj )Fj ,

j=r+1

Then y(x) = ϕ1 (x) + ϕ2 (x) + ϕ3 (x) + ϕ4 (x). On the other hand, for all (0 0, and therefore, as the numbers si and si+1 are arbitrary, we obtain condition 1◦ . Conversely, from the conditions 1◦ , 2◦ and 3◦ of Definition 3, by virtue of the same criterion of oscillatory character, follows that the matrix K(si , sk )n1 . is oscillatory. 7. Influence function of a string Let S now denote a string fastened at the points 0 and l with tension T . The line of deflection of the string under the influence of n concentrated forces F1 , . . . , Fn will be a broken line with ends at the points 0 and l of the X axis, and with vertices on the lines of action of the forces F1 , . . . , Fn (Figure 1).

7. INFLUENCE FUNCTION OF A STRING

s

147

l−s

α

β

x

f

y Figure 2 Therefore in this case properties A and B hold, and consequently one can state that the influence function of the string is an oscillatory function. This can also be verified by direct calculation, by setting up the expression for K(x, s). Indeed, under the influence of a unit force, the line of deflection of the string assumes the shape indicated in Figure 2. Let us abbreviate K(s, s) by f . Since the tensions T at the point C should balance the unit force F , we obtain from the equations for the projections of the forces on the Y axis T (sin α + sin β) = 1. Since the angles α and β are small, we can put here sin α =

f , s

We then find

sin β =

f . l−s

1 (l − s)s. Tl From the similarity of the triangles we obtain two different expressions for the deflection at a point x, depending on whether x < s or x > s: ⎧ x(l − s) ⎪ ⎨ (x ≤ s), Tl K(x, s) = (69) ⎪ ⎩ s(l − x) (x ≥ s). Tl Let us now take an arbitrary system of numbers (0 0, χi > 0 (i = 1, 2, . . . , n) and

ψ1 ψ2 ψn < < ··· < . χ1 χ2 χn

By Theorem 12 of Chap. II the matrix (70) is oscillatory, q.e.d. We have assumed that both ends of the string are stationary. If one of the ends, for example, the second, can freely slide along a line parallel to the Y axis, then the oscillatory character of the influence function follows from analogous considerations. It is easy to obtain for this case the following expression for the influence function: ⎧ x ⎨ (x ≤ s), T K(x, s) = (71) ⎩ s (x ≥ s). T If, to the contrary, the first end slides and the second one is stationary, then ⎧ l−s ⎪ ⎨ (x ≤ s), T (72) K(x, s) = ⎪ ⎩ l − x (x ≥ s). T n The matrix K(xi , xk )1 is again a single-pair oscillatory matrix. We have thus established: Theorem 3. The influence function of a string under ordinary conditions of fastening of the ends is oscillatory. 8. Influence function of a rod 1. In the present section we shall prove that the influence function of a rod S is oscillatory, for all conditions of fastening encountered in engineering. The proof will consist in verifying property B of the rod S. In this verification we shall use some refinements of Rolle’s theorem, with which we shall begin. Let ϕ(x) be a continuous function in a closed interval [a, b]. We consider the set of all zeros of this function. This set will consist, generally speaking, of intervals and of points not belonging to any one of these intervals. Each of the intervals, and also each of the individual points, will be called a zero place of the function ϕ(x). A zero place of a function ϕ(x) will be called nodal if upon passing through it the function ϕ(x) reverses sign15 . Thus, by the definition, a nodal zero place should lie with all its points in the interior of the interval [a, b]. It is also obvious that the number of nodal places of a function ϕ(x) which reverses sign a finite number of times coincides with the number of sign changes of the function ϕ(x) on the interval [a, b]. It is easy to see that for functions ϕ(x) which have continuous derivatives the following refinement of Rolle’s theorem holds: 15 That is, in any neighborhood of this zero place there are two points, lying on opposite sides of it, in which the function ϕ(x) takes values with opposite signs.

8. INFLUENCE FUNCTION OF A ROD

149

1◦ Between each two zero places of the function ϕ(x) there is at least one nodal place of its derivative ϕ (x). By this theorem, if the derivative ϕ (x) has n different nodal places, then the function ϕ(x) has at most n + 1 zero places. Let us prove now the following proposition: 2◦ If the derivative ϕ (x) is continuous in the closed interval [a, b] and has in it n nodal places, then the function ϕ(x) has in the closed interval [a, b] at most n zero places, if at least one of the following conditions is satisfied ϕ(a)ϕ (a) > 0,

ϕ(b)ϕ (b) < 0,

(73)

and at most n − 1 zero places, if both of these conditions are satisfied. Indeed, let, for example, the first inequality (73) be satisfied with ϕ(a) > 0

and

ϕ (a) > 0.

Let α be the nodal zero16 of the function ϕ (x) closest to a. Then ϕ (x) ≥ 0 on the interval (a, α) and consequently, the function ϕ(x) does not decrease in this interval, so it remains greater than ϕ(a) > 0. On the other hand, on the interval (α, b), the function ϕ (x) obviously has n − 1 nodal places; consequently, ϕ(x) has, according to 1◦ , at most n zero places in the closed interval [α, b], and thus also on the interval [a, b]. The cases when the other condition (73) is satisfied or when both of these conditions are simultaneously satisfied are analyzed similarly. 2. Let us proceed now to a direct examination of the problem of interest to us. We shall start with the well-known differential equation of bending from the theory of elasticity: (EIy  ) = q(x), (74) where EI is the so-called stiffness of the rod (generally speaking, depending on x), and q(x)dx is the load on an element dx of the rod. In addition, in this equation we assume that the functions y(x), y  (x), and EIy  are continuous, and assume that the quasi-derivative (EIy  ) is continuous everywhere except at the points where the concentrated forces are applied; namely, if the concentrated forces Fi are applied at the points si , then (EIy  )x=si +0 − (EIy  )x=si −0 = Fi

(i = 1, 2, . . . , n).

(75)

Term-by-term integration of (74) from zero to x, taking into the account (75), results in (EIy  ) = Q(x), (76) where Q(x) is the so-called shearing force at the section x (the sum of all forces applied to the left of the section x):  x  Q(x) = q(s)ds + R0 + Fi 0

si 0, to which corresponds the orthonormal system of eigenvectors v i = (v1i , . . . , vni ) n  vji vjk = δik (v i , v k ) =

(i = 1, 2, . . . , n), (i, k = 1, 2, . . . , n).

j=1

Introducing with the aid of (109) the corresponding vectors ui (i = 1, 2, . . . , n),  we obtain a complete system of eigenvectors of the matrix A:  i = λ i ui Au

(i = 1, 2, . . . , n).

(110)

Here the vectors ui = (u1i , . . . , uni ) (i = 1, 2, . . . , n) are orthonormal with respect to the system of masses mi , . . . , mn ; i.e. m 

mj uji ujk = δik

(i, k = 1, 2, . . . , n).

(111)

j=1

Comparing (110) with (108) we conclude that in our system Sn there exist n natural oscillations with frequencies 1 pi = √ λi

(i = 1, 2, . . . , n)

(112)

and corresponding amplitude vectors ui (i = 1, 2, . . . , n) for which the conditions of the generalized orthonormalization (111) are satisfied. We shall denote by the letter y an n-dimensional vector (y1 , y2 , . . . , yn ). Then the system of (104) can be written in the form of a single vector equation: y. y = −A¨

(113)

9. SMALL OSCILLATIONS OF AN ELASTIC CONTINUUM WITH n MASSES

159

Repeating essentially the considerations from Section 1, let us show that an arbitrary oscillation of the system is obtained by superposition of the individual natural oscillations; i.e., y=

n 

Ci ui sin(pi t + αi ),

i=1

or y=

n 

(Ai cos pi t + Bi sin pi t)ui .

(114)

i=1

Indeed, expanding the vector y with respect to ui (i = 1, 2, . . . , n), we obtain y=

n 

θ i ui

(θi = θi (t); i = 1, 2, . . . , n),

(115)

i=1

and then we have, by (115), (110) and (112): ¨ y=

n 

y = A¨

θ¨i ui ,

i=1

n 

 i= θ¨i Au

i=1

n ¨  θi i u. p2 i=1 i

Thus, (113) is equivalent to the system of equations θ¨i + p2i θi = 0

(i = 1, 2, . . . , n).

From these equations, we obtain θi = Ai cos pi t + Bi sin pi t

(= ci sin(pi t + αi )),

where Ai and Bi (i = 1, 2, . . . , n) are arbitrary constants. Inserting the expressions obtained for θi (i = 1, 2, . . . , n) into (115) we get (114). Let us show also how one determines the arbitrary constants from the initial data ˙ y(0) = (y10 , . . . , yn0 ), y(0) = (y˙ 10 , . . . , y˙ n0 ). By (111), it follows from (114) that Ai cos pi t + Bi sin pi t =

n 

mj uji yj ,

(116)

j=1

therefore by differentiating, we obtain pi (−Ai sin pi t + Bi cos pi t) =

n 

mj uji y˙ j .

(117)

j=1

Putting t = 0 in (116) and (117), we obtain Ai =

n  j=1

mj uji yj0 ,

Bi =

n 1  mj uji y˙j0 pi j=1

(i = 1, 2, . . . , n).

Thus, the motion of the system of masses m1 , . . . , mn has been expressed in a closed from. To determine the motion of any point of the continuum it remains to insert the expressions obtained for the yk as functions of time into the right hand side of (103).

160

III. SYSTEMS WITH n DEGREES OF FREEDOM

In concluding this section, let us write out in greater detail the equation from which we determine the square of the frequency λ = p2 of a free oscillation. From (117) we obtain |δik − λαik mk |n1 = 0. The polynomial Δ(λ) in the left side of this equation has the following expansion: Δ(λ) =

n 

cp (−λ)p

(c0 = 1),

(118)

p=0

   αi i . . . αi i  1 p   11  . . . . . . . . . . . . . . . .  mi . . . mi . cp = (119) 1 p   i1 · · · > λn > 0. The amplitude vectors uj (j = 1, 2, . . . , n) of the natural oscillations of the system Sn are the eigenvectors of the matrix A, and the corresponding frequencies p1 , . . . , pn are obtained from (112); hence by recalling the spectral properties of the oscillatory matrices (Section 5 of Chap. II) we arrive at the following theorem: Theorem 6. A segmental continuum loaded with n concentrated masses has n different frequencies p1 < p2 < · · · < pn . In the case of a harmonic oscillation with frequency pj (j = 1, 2, . . . , n), the amplitude deflections u1 , u2 , . . . , un at the points of application of the masses m1 , . . . , mn form a sequence which has exactly j − 1 sign changes. Let us investigate now the form of the line of deflection under harmonic oscillation with frequency p. Since in this case yk = uk sin(pt + α) (k = 1, 2, . . . , n), we have that d2 yk = −p2 uk sin(pt + α) (k = 1, 2, . . . , n). dt2 19 When

n = 2 we assume that at least one of the points s1 or s2 is interior.

10. SMALL OSCILLATIONS OF A SEGMENTAL CONTINUUM

161

Therefore formula (103) for the harmonic oscillation under consideration will yield y(x, t) = u(x) sin(pt + α),

(121)

where the deflection amplitude u(x) is determined from the formula u(x) = p2

n 

K(x, sk )mk uk .

(122)

k=1

Since K(x, s) > 0 when 0 < x < l; s ∈ I, we obtain from formula (122) the first part of the following theorem: Theorem 7. 1. In a segmental continuum loaded with n concentrated masses the amplitude line u(x) of the fundamental tone (p = p1 ) does not intersect the X axis on the interval I. 2. The amplitude line of the j-th overtone (p = pj+1 ) has exactly j nodes and no other intersections with the X axis on the interval I. To prove the second part of the theorem we note that by Theorem 6, there are exactly j sign changes for the j-th overtone in the sequence of numbers u(s1 ), u(s2 ), . . . , u(sn ). Let, for example, uk1 = u(sk1 ) > 0, uk2 = u(sk2 ) < 0, . . . . . . , (−1)j ukj+1 = (−1)j u(skj+1 ) > 0. Then between each two adjacent points of the sequence sk1 , . . . , skj+1 there is at least one nodal zero20 of the deflection amplitude u(x). We denote these nodal zeros by x1 , . . . , xj ; then ⎫ u(xg ) = 0 (g = 1, 2, . . . , j) ⎪ ⎬ and (123) ⎪ ⎭ sk1 < x1 < sk2 < x2 < · · · < xj < skj+1 . Let us assume now that the function u(x) vanishes also in a movable point different from x1 , . . . , xj ; let, for example, u(x∗ ) = 0,

(124)

where

xh ≤ x∗ ≤ xh+1 (0 ≤ h ≤ j; x0 = 0; xj+1 = l). (125)   ∗ We denote the numbers x1 , . . . , xj , x , arranged in increasing order, by x1 < x2 < · · · < xj+1 and insert successively these numbers for x in (122); we have n 

K(xi , sk )mk uk = 0 (i = 1, 2, . . . , j + 1).

k=1

This is a system of j + 1 homogeneous linear equations with respect to the variables mk uk (k = 1, 2, . . . , n), and the matrix of the coefficients of this system has rank j + 1 since by (123) and (125), x1 , sk1 < x2 , sk2 < · · · < xj+1 , skj+1 , 20 See

footnote on p. 150.

162

III. SYSTEMS WITH n DEGREES OF FREEDOM

and therefore, on the basis of the remark made in the end of Section 8,   x1 x2 . . . xj+1 = 0. K sk1 sk2 . . . skj+1

(126)

Since the number of sign changes in the sequence u1 , u2 , . . . , un , is exactly j, these numbers can be broken up into j + 1 groups u1 , . . . , uν1 ; uν1 +1 , . . . , uν2 ; . . . ; uνj +1 , . . . , uνj+1

(νj+1 = n),

such that the members of each group are of the same sign, uk = εh |uk |,

νh−1 < k ≤ νh

(h = 1, 2, . . . , j + 1; ν0 = 0, νj+1 = h),

and in each group there are numbers different from zero. Let us put Aih =

νh 

K(xi , skμ )mμ |uμ |

(i, h = 1, 2, . . . , j + 1).

(127)

μ=νh+1 +1

Then

j+1 

Aih εh = 0 (i = 1, 2, . . . , j + 1)

h=1

and, consequently,

 1 2 ... j + 1 = 0. A 1 2 ... j + 1 But this is impossible since, by (126) and (127),   1 2 ... j + 1 = A 1 2 ... j + 1   x1 = mμ1 μμ2 . . . mμj+1 |uμ1 uμ2 . . . uμj+1 |K sk μ 1 

1≤μ1 ≤ν1 ν1 0.

Thus (124) has produced a contradiction and the theorem is proved. Theorem 6 of Sec. 5, Chap. II makes it possible to prove the following: Theorem 8. If a free oscillation of a segmental continuum loaded with n concentrated masses is obtained by superposition of natural oscillations with frequencies pk < pl < · · · < pn , then the number of sign changes in the line of deflection fluctuates between k − 1 and m − 1 during the motion. Proof. We again introduce the vector y = (y1 , y2 , . . . , yn ). For the oscillation under consideration y = Ck sin(pk t + αk )uk + Cl sin(pl t + αl )ul + · · · + Cm sin(pm t + αm )um ; hence, by Theorem 6 of Chap. II, the number of sign changes in the sequence of coordinates of the vector y as well as in the sequence of coordinates of the vector ¨ y = −p2k Ck sin(pk t + αk )uk − · · · − p2m Cm sin(pm t + αm )um , fluctuates from k − 1 to m − 1. Since yi = y(si , t) (i = 1, 2, . . . , n),

11. OSCILLATIONS OF CONCENTRATED MASSES ON A BEAM

163

we conclude that at any instant t the number of sign changes of the line of deflection y(x, t) is at least k − 1. On the other hand, since the d’Alembert principle (see formula (103)) applies, it is possible to consider the deflection y(x, t) at any instant of time t as a static deflection, caused by the inertia forces Fi = −mi y¨i

(i = 1, 2, . . . , n);

hence by the strengthened property B of the segmental continuum (see Sec. 6.3), the number of sign changes in the line of deflection y(x, t) does not exceed the number of sign changes in the sequence F1 , . . . , Fn . Since the number of sign changes in the sequence F1 , . . . , Fn coincides with the number of sign changes in the sequence y¨1 , . . . , y¨n , and consequently does not exceed m − 1, the deflection y(x, t) changes sign at most m − 1 times. This proves the theorem. In Chap. IV (Section 4, Theorem 5) we shall prove that for an arbitrary distribution of masses along the continuum (and thus also in the case of m concentrated masses), the nodes of the amplitude lines of two adjacent overtones alternate. We shall also prove there the following: If the free oscillation of a segmental continuum is obtained by superposition of natural oscillation with frequencies pk < pl < · · · < pm , then at any instant of time the line of deflection has at least k − 1 nodal places and at most m − 1 zero places in I. 11. Oscillations of a system of concentrated masses placed on a multi-span beam Let S ∗ be a beam with p + 1 spans, the intermediate supports of which are located at the points c1 < c2 < · · · < cp . Assume that at the movable points s1 < s2 < · · · < sn different from the support points of the beam S ∗ , n masses are concentrated, m1 , . . . , mn . Neglecting the mass of the beam, let us investigate small free oscillations of this system. We first assume that the conditions of fastening at the ends are such that when the intermediate supports are removed the beam cannot move as a solid body (there are no cantilever ends or, if one end is cantilevered, then the second end is rigidly fastened, etc.). We denote by K(x, s) (0 ≤ x, s ≤ l) the influence function of the single-span beam S obtained from S ∗ by removing the intermediate supports at the points c1 , c2 , . . . , cp . Then, as was seen in Section 4, the influence function K ∗ (x, s) of the beam S ∗ will be   x c1 . . . cp K s c1 . . . cp   (128) K ∗ (x, s) = c 1 . . . cp K c1 . . . cp We denote by yi the deflection of the beam at the point si (i = 1, 2, . . . , n), and obtain for y1 , y2 , . . . , yn the system of ordinary differential equations (see Section 9): yi = −

n  k=1

∗ αik

d2 yk dt2

(i = 1, 2, . . . , n),

(129)

164

III. SYSTEMS WITH n DEGREES OF FREEDOM

where

∗ = K ∗ (si , sk )mk (i, k = 1, 2, . . . , n). αik ∗ n By (128), the matrix αik 1 , unlike that analyzed in Section 10 for the case of a single-span beam (or string), is no longer oscillatory. So we have to transform the system (129). The points c1 , c2 , . . . , cp divide the beam into p + 1 spans. Let us put εi (i = 1, 2, . . . , n) equal to +1 or −1, depending on whether si belongs to an odd or to an even span, and make a change of variables

yi = εi yi

(i = 1, 2, . . . , n).

Then the system (129) assumes the form yi = −

n 

α ˜ ik

d2 y˜k dt2

si sk 

c1 c1

k=1

where

 K

∗ α ik = εi εk αik = εi εk mk

K

c1 c1

(i = 1, 2, . . . , n),

. . . cp . . . cp  . . . cp . . . cp

(130)

 (i, k = 1, 2, . . . , n)

(131)

In the system (130), the matrix  αik n1 is oscillatory. Indeed, since K(x, s) is an oscillatory function, then by Sylvester’s identity and the choice of signs of εi (i = 1, 2, . . . , n), we have:    i1 . . . iq A k1 . . . kq   s i1 . . . s iq c 1 . . . c p K sk 1 . . . sk q c1 . . . cp   ≥ 0, = εi1 . . . εiq εk1 . . . εkq mk1 . . . mkq c 1 . . . cp K c1 . . . cp   i1 < · · · < iq 1≤ ≤n , k1 < · · · < kq with strict inequality if i1 = k1 , . . . , iq = kq . In addition, according to Theorem 5 of Sec. 8, Chap. III, α ik > 0 (i, k = 1, 2, . . . , n). (132) In view of the matrix  αik n1 of the transformed system (130) being oscillatory, all the results of the preceding section remain in force if we replace in them the deflections yi by yi = εi yi (i = 1, 2, . . . , n), the amplitude deflections ui by u i = εi ui (i = 1, 2, . . . , n) and the amplitude line u(x) by u (x) = ε(x)u(x), where ε(x) is equal to +1 or −1, depending on whether the point x belongs to an odd or to an even span. In this case Theorem 7 of Section 10 should be reformulated as follows: The function u(x) = ε(x)u(x), where u(x) is the amplitude function of the j-th harmonic oscillation, has j − 1 nodes (some of which may fall on the supports) and has no other zeros between the supports.

11. OSCILLATIONS OF CONCENTRATED MASSES ON A BEAM

165

We call the reader’s attention to the following fact. Since Theorem 6 of Section 10 remains in force here, independent of the arrangements of the supports and masses, the frequencies of all harmonic oscillations are all distinct. In these considerations we have excluded all cases when the fastenings at the ends are such that after the removal of the intermediate supports the beam can move as a rigid body. However, all excluded cases can be treated as limiting cases with respect to those analyzed here. Therefore, in these cases the matrix  αik  will be totally non-negative. Since |Lik |n1 = |αik |n1 > 0 and (132) holds, then the matrix  αik n1 is oscillatory, and all the results of this section remain in force.

http://dx.doi.org/10.1090/chel/345/05

CHAPTER IV

Small Oscillations of Mechanical Systems with an Infinite Number of Degrees of Freedom In Chapter III we studied oscillations of a massless linear elastic continuum with n concentrated masses. The results obtained there can be extended in a natural manner to include the general case of oscillations of a linear elastic continuum which carries both concentrated masses and masses that are distributed continuously. Here it is necessary merely to replace the system of ordinary linear differential equations which represents the oscillations by one integro-differential equation, and to replace the system of algebraic equations from which the deflection amplitudes at the points of concentration of the masses are determined by linear integral equations for the amplitude functions. Assuming now that the reader is familiar with the theory of linear integral equations (at least those with symmetric kernel)1 , let us investigate the oscillations of a linear elastic continuum with an arbitrary mass distribution; we do not limit our attention to free oscillations, but also consider forced oscillations.

1. Principal premises 1. The integro-differential equation for the oscillations. Let S denote the same linear elastic continuum as in Section 4 of Chap. III, and let K(x, s) be its influence function. If external forces are applied to the continuum, then by the principle of independence of the action of the forces the deflection y(x) is given by the formula  y(x) = K(x, s)dQ(s), (1) S

where dQ(s) is the force applied to the section ds and the integral is taken over the entire system. In formula (1) the integral is taken in the sense of Stieltjes2 , and therefore this formula, which is the generalization of formula (47) of Chap. III, is suitable for the most general case, when there are both continuous and concentrated forces. 1 See

I.G. Petrovsky [41] and I.I. Privalov [43], and also Courant–Hilbert [8] and S.G. Mikhlin

[39]. 2 A treatment of the elements of the theory of Stieltjes integrals can be found in the book by I.P. Natanson, Teoriya funktsii veschestvennoy peremennoy (Theory of Functions of Real Variable), Gostekhizdat, 1950. More detailed information on the Stieltjes integral can be found in the book by V.I. Glivenko, Integral Stil’t’esa (The Stieltjes Integral), ONTI, 1936. See also L.V. Kantorovich, On the Theory of the Stieltjes–Riemann Integral, Uchenye zapiski Leningradskogo universiteta (Scientific Notes of the Leningrad University), No. 37, (1939), pp. 52–68.

167

168

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

Analogously, formula (51) of Chap. III for the energy V stored by the continuum under the influence of n concentrated forces can be extended to the general case:   1 V = K(x, s)dQ(x)dQ(s). (2) 2 S S Suppose now that time-dependent disturbing forces are applied to the continuum S: dQ(s) = dQ(s, t). We use the d’Alembert principle to write the equations of motion and replace dQ(s) in (1) by dQ(s, t) −

∂2y dσ(s), ∂t2

where dσ(s) is the mass per element ds of the continuum. We obtain the following integro-differential equation:   ∂ 2 y(s, t) y(x, t) = − K(x, s) dσ(s) + K(x, s)dQ(s, t). (3) ∂t2 S S In the absence of external forces (3) becomes simpler:  ∂ 2 y(s, t) dσ(s). y(x, t) = − K(x, s) ∂t2 S

(3 )

2. Natural Oscillations. Before proceeding to solve (3) or (3 ) under certain initial conditions, let us first investigate the natural harmonic oscillations of the continuum S, i.e., oscillations of the form y(x, t) = ϕ(x) sin(pt + α), where ϕ(x) is the amplitude function and p the frequency of oscillation. Substituting this expression for y(x, t) in (3 ) we get  ϕ(x) = λ K(x, s)ϕ(s)dσ(s) (λ = p2 ).

(4)

S

We see that the squares of the unknown frequencies are the eigenvalues of the integral equation (4), and the unknown amplitude functions are the corresponding fundamental functions. We know (see Chap. III, p. 155) from the Maxwell reciprocity principle that K(x, s) is a symmetric kernel. Since (4) contains dσ(s) and not ds, (4) is a loaded integral equation with symmetric kernel. One can apply to a loaded integral equation the entire theory of ordinary integral equations if one substitutes the usual differentials ds, dr, . . . by the corresponding Stieltjes differentials dσ(s), dσ(r), . . . . Thus, for example, the eigenvalues λi (i = 1, 2, . . . ) of (4) are the roots of the entire transcendental function (called the Fredholm determinant) D(λ) =

∞ 

(−λ)k ck ,

(5)

k=0

where n

       1 s1 . . . sn dσ(s1 ) . . . dσ(sn ) (n = 1, 2, . . . ). ··· K c0 = 1, cn = s1 . . . sn n! S S

(5 )

1. PRINCIPAL PREMISES

169

The equality n

       ∞  (−λ)n x s1 . . . sn dσ(s1 ) . . . dσ(sn ) (6) ··· K D(x, s; λ) = K(x, s) + s s2 . . . sn n! S S n=1 defines an entire transcendental function (with respect to λ), which is called the Fredholm minor. If λ is not an eigenvalue of (4), then the non-homogeneous equation  b ϕ(x) = λ K(x, s)ϕ(s)dσ(s) + f (x), (7) a

has a unique solution, determined by the formula  b Γ(x, s; λ)f (s)dσ(s), ϕ(x) = f (x) + λ

(8)

a

where

D(x, s; λ) (9) D(λ) is the resolvent of the integral equation (7). At values of λ of sufficiently small moduli, the resolvent is represented by a power series that converges uniformly with respect to x and s Γ(x, s; λ) =

Γ(x, s; λ) = K(x, s) + λK (2) (x, s) + λ2 K (3) (x, s) + . . . , where n−1

     K (n) (x, s) = · · · K(x, s1 )K(s1 , s2 ) . . . K(sn−1 , s)dσ(s1 ) . . . dσ(sn−1 ) S

S

(n = 2, 3, . . . ) is the n-th iterated kernel for the kernel K(x, s). From the power series representation follows the functional equation for the resolvent:  b

Γ(x, s; λ) = K(x, s) + λ

Γ(x, t; λ)K(t, s)dσ(t).

(10)

a

If the kernel is symmetric, then the numbers λi (i = 1, . . . , n) are all real and to the sequence of eigenvalues {λi } (where each eigenvalue is listed according to its multiplicity3 , one can put into correspondence a sequence of fundamental functions {ϕi (x)}:  ϕi (x) = λi

K(x, s)ϕi (s)dσ(s) (i = 1, 2, . . . )

(11)

S

which is orthonormal with respect to the differential dσ:   1 (i = k) ϕi (x)ϕk (x)dσ(x) = (i, k = 1, 2, . . . ). 0 (i = k) S

(12)

Every other fundamental function which does not enter into the sequence {ϕi (x)}, is expressed as a linear combination of a finite number of functions of this sequence. 3 As

the root of the equation D(λ) = 0.

170

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

Since the double integral (2) is always positive by its mechanical meaning (provided dQ(s) = 0 for at least one movable point s)4 , the kernel K(x, s) is positive definite in the strong sense5 on the set I of the movable points of the continuum S; i.e.   K(x, s)dQ(x)dQ(s) > 0 (13) S

whenever

S

 |dQ(s)| > 0. I

Therefore, all λi > 0 (i = 1, 2, . . . ), which incidentally can be proved by the relations: $ 2   ϕ (x)dσ(x) 1 = S i = K(x, s)ϕi (x)ϕi (s)dσ(x)dσ(s) > 0 λi λi S S (i = 1, 2, . . . ), which follow from (11), (12) and (13) with dQ(s) = ϕi dσ(s). Since the frequencies pi (i = 1, 2, . . . ) of the harmonic oscillations y = ϕi (x) sin(pi t + αi )

(i = 1, 2, . . . ),

(14)

which form a complete system of linearly independent harmonic oscillations are obtained from the formulas pi = λi (i = 1, 2, . . . ), we conclude from the positivity of all the λi , that the pi are real and consequently the oscillations (14) are stable harmonic oscillations, as expected. For symmetric positive definite integral kernels, according to the Mercer theorem from the theory of ordinary integral equations, we have the bilinear expansion K(x, s) =

∞  ϕi (x)ϕi (s) , λi i=1

(15)

which converges absolutely and uniformly in the region where the kernel K(x, s) is defined. For loaded integral equations, the Mercer theorem remains valid, provided that dσ(s) = 0 for every movable point s. In the general case, when S contains segments that have no masses, the expansion (15) holds (furthermore, absolutely and uniformly) for all x and for all numbers s for which dσ(s) = 0. For a positive definite kernel, the Fredholm determinant has the form  ∞   λ D(λ) = 1− . (16) λi i=1 4 i.e.,

there exists no neighborhood of the point s in which dQ(s) vanishes identically. one means by a positive definite kernel K(x, s) (if it is continuous) a symmetric kernel for which the sign ≥ stands instead of > in (13). In our case the strong condition of positive definiteness is satisfied. From the strong positive definiteness, by a theorem by F . Riesz (see S. Banach [2], pp. 48–49), it follows that every continuous function ϕ(s) which vanishes at the stationary supports, can be uniformly approximated by linear combinations of the fundamental functions ϕi (x). 5 Frequently

1. PRINCIPAL PREMISES

171

We have already noticed that formulas (1) and (2) for the deflection and for potential energy are generalizations of the corresponding formulas of Section 4 of Chap. III. We call attention also to the fact that the equation Δ(λ) = 0 in Section 9 of Chap. III can be obtained as a particular case of the equation D(λ) = 0 when dσ(s) differs from 0 only at n points s1 , s2 , . . . , sn . On the other hand, the reader will readily observe, by comparing formulas (118) and (120) (Section 9, Chap. III) with formula (5), that D(λ) can be obtained from Δ(λ) by passing to the limit. We also notice that in the case of a continuous symmetric kernel, the eigenvalues of a loaded integral equation (as of an ordinary one) have the minimax properties6 . We shall henceforth make use of only the maximal property of the smallest positive eigenvalue. The integral form 

b



b

K(x, s)ϕ(x)ϕ(s)dσ(x)dσ(s), a

a

if it assumes any positive values, assumes a maximum value on the “unit sphere” 

b

ϕ2 (x)dσ(x) = 1, a

This maximum value is 1/λ0 , where λ0 is the smallest positive eigenvalue of (6) and any normalized function ϕ(x) for which this maximum value is attained is a fundamental function of (6) for λ = λ0 . 3. Solution of the integro-differential equation. After these preliminary remarks, let us proceed to the investigation of the integro-differential equation (3) of forced oscillations. Here, without loss of generality, we can confine ourselves to the case that the forces are applied only to such points and sections where there are masses; in other words, we can assume that if there are no masses in a certain section, then no disturbing forces are applied there. Indeed, were we to have forces applied to sections free of mass, we could isolate these forces into one group and find the deflection y1 (x, t) for the oscillation produced by this group of forces, using the static formula  y1 (x, t) = K(x, s)dQ(s, t), S1

where S1 is that portion of S on which there are no masses. After having then found the deflection y2 (x, t) for the oscillation produced by the remaining group of forces, we obtain the required deflection y(x, t) of the total oscillation by adding y1 (x, t) to y2 (x, t) (by the principle of independence of the action of forces). Thus, assuming dQ(s, t) = 0 when dσ(s) = 0, we insert (15) into (3). Since the expansion (15) converges uniformly with respect to x on the set of s on which dσ(s) = 0, such substitution is legitimate. After the substitution we obtain y(x, t) =

∞ 

Ti (t)ϕi (x),

(17)

i=1 6 These properties are analogous to the minimax properties of the eigenvalues of quadratic forms, discussed in Chap. I, Section 10; see Courant–Hilbert [8], Vol. I, pp. 122, 383.

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IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

where

 ∂ 2 y(s, t) 1 dσ(s) + ϕi (s)dQ(s, t) ∂t2 λi S (17 ) S (i = 1, 2, . . . ). Thus, the deflection y(x, t) is expandable into a uniformly convergent series in the fundamental functions ϕi (x) (i = 1, 2, . . . ). As the system of fundamental functions ϕi (x) (i = 1, 2, . . . ) is orthonormal, we obtain from (17) that  Ti (t) = −

1 λi



ϕi (s)

Ti (t) =

y(x, t)ϕi (x)dσ(x);

(18)

S

hence



∂ 2 y(s, t) ϕi (x)dσ(x). ∂t2 S Comparing these equations with (17 ) we obtain the following differential equations: T¨i + λi T = fi (t) (i = 1, 2, . . . ), (19) T¨i (t) =



where fi (t) =

ϕi (s)dQ(s, t) (i = 1, 2, . . . ). S

From (19) we obtain Ti (t) = Ai cos pi t + Bi sin pi t + Ti0 (t)

(pi =

λi ; i = 1, 2, . . . ),

(20)

where Ai and Bi (i = 1, 2, . . . ) are arbitrary constants, and Ti0 (t) (i = 1, 2, . . . ) is a particular solution to (19). It is sometimes convenient to use the particular solution  t 1 Ti0 (t) = sin pi (t − τ )fi (τ )dτ, pi 0 which satisfies the zero initial conditions: Ti0 (0) = T˙i0 (0) = 0

(21)

Thus, we finally obtain the following expression for the sought displacement y(x, t): ∞  y(x, t) = (Ai cos pi t + Bi sin pi t + Ti0 (t))ϕi (x) (22) i=1

In order to determine the arbitrary constants, it is necessary to specify some other data. Usually one specifies the initial data, i.e., the initial displacement y(x, 0) = η0 (x)

(23)

y(x, ˙ 0) = η1 (x).

(23 )

and the initial velocity To determine the constants Ai and Bi (i = 1, 2, . . . , n) from the given functions η0 (x) and η1 (x), we proceed in the following manner. It follows from (18) and (20) that  Ai cos pi t + Bi sin pi t + Ti0 (t) = y(x, t)ϕi (x)dσ(x) (i = 1, 2, . . . ). S

1. PRINCIPAL PREMISES

173

Putting here t = 0, we determine Ai (i = 1, 2, . . . ); differentiating both sides of the preceding equation with respect to t and putting again t = 0, we determine Bi (i = 1, 2, . . . ). Thus, taking (21) into consideration, we find  ⎫ ⎪ η0 (x)ϕi (x)dσ(x), ⎪ Ai = ⎬ S  (i = 1, 2, . . . ), (24) 1 ⎪ ⎭ Bi = η1 (x)ϕi (x)dσ(x)⎪ pi S and this concludes the solution to the integro-differential equation (3) with the initial conditions (23) and (23 ). The foregoing derivation of (22) constitutes the justification of the so-called Fourier method, which in its ordinary form contains certain elements of arbitrariness, for it leaves open the question of the existence of the fundamental functions, and requires an ` a posteriori justification for each special case. Let us indicate furthermore that the Fourier method in the form developed and used by its leading practitioners — Fourier, Poisson, and later Poincar´e — had as its starting point a differential equation of the oscillation. However, the integro-differential equation (3) cannot always be replaced by an equivalent partial differential equation with boundary conditions (and, in the presence of concentrated masses or forces, with additional discontinuity conditions). For this it is necessary that the distribution of masses and the disturbing forces satisfy additional conditions (the existence of density, the absence of an infinite number of concentrated masses, etc.). It must be emphasized that our investigation of the integro-differential equation (3) is not complete from the mathematical point of view. Indeed, all our preceding analysis has only shown that if there exists a solution to the integro-differential equation (3) subject to the initial conditions (23) and (23 ), then it is unique and is expressed by uniformly convergent series (23), where the Ai and Bi (i = 1, 2, . . . ) are obtained from (24). One can raise the following question: what should the functions η0 (x), η1 (x), and the disturbing forces dQ(s) be in order for a solution to (3), satisfying the initial conditions (23) and (23 ), to exist (here, naturally, one must make more precise: to ∂2y what class of functions the functions should the derivatives 2 belong)? It follows ∂t from (22) with t = 0 that ∞  η0 (x) = Ai ϕi (x). i=1

This in itself already shows that we cannot choose for η0 (x) any continuous function of x, but only such a function for which the generalized Fourier expansion with respect to the fundamental functions ϕi (x) (i = 1, 2, . . . ) converges uniformly. In addition to this condition, η0 (x), η1 (x) and the differential dQ(s, t) should satisfy several other conditions which will guarantee that the function y(x, t), determined from (22) has a second derivative. All these questions have not yet been investigated sufficiently from our point of view, and we shall not touch them. 4. Case of pulsating external forces. Let us dwell in greater detail on the case that the system of disturbing forces has a pulsating character (i.e., depends

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IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

harmonically on the time), which can be written mathematically as dQ(s, t) = sin(pt + α)dQ(s). Equation (3) can now be rewritten as  ∂ 2 y(s, t) K(x, s) dσ(s) + F (x) sin(pt + α), y(x, t) = − ∂t2 S

(25)



where F (x) =

K(x, s)dQ(s).

(26)

S

Let us try to obtain a particular solution to (25) in the form y(x, t) = u(x) sin(pt + α).

(27)

After substitution in (25) and dividing by sin(pt + α), we obtain the following equation for u(x):  K(x, s)u(s)dσ(s). (28) u(x) = F (x) + p2 S

Assuming that the frequency p does not coincide with any of the natural frequencies, and consequently p2 is not an eigenvalue of the integral equation (28), we obtain the solution u(x) of this equation from the formula:  2 u(x) = F (x) + p Γ(x, s; p2 )F (s)dσ(s), (29) S

where Γ (x, s; λ) is the resolvent of the loaded integral equation (28), and consequently, satisfies the relation  Γ(x, s; λ) = K(x, s) + λ Γ(x, r; λ)K(r, s)dσ(r). (30) S

Inserting into (29) the expression for F (x) from (26) and using the identity (30) for the resolvent, we readily obtain after elementary transformations  u(x) = Γ(x, s; p2 )dQ(s). (31) S

If we use the well-known expansion of the resolvent of a positive definite kernel Γ(x, s; λ) =

∞  ϕi (x)ϕi (s) , λi − λ i=1

(32)

which, in this case, converges absolutely and uniformly for all x and for those s for which dσ(s) = 0, we obtain also the following expression for u(x): u(x) =

∞  ci ϕi (x) , p2i − p2 i=1

(33)



where ci =

ϕi (s)dQ(s) S

(i = 1, 2, . . . ).

(34)

1. PRINCIPAL PREMISES

175

It is obvious that the general solution to the linear integro-differential equation (25) can, by its linearity, be represented in the form y(x, t) = u(x) sin(pt + α) + Y (x, t),

(35)

where Y (x, t) is the solution to the corresponding homogeneous equation and consequently has the form Y (x, t) =

∞ 

(Ai cos pi t + Bi sin pi t)ϕi (x).

i=1

To determine the arbitrary constants Ai and Bi (i = 1, 2, . . . ), from the initial data (22) and (23), it must be noted that  ⎫ ⎪ ⎪ Ai = Y (x, 0)ϕi (x)dσ(s), ⎬ S  (i = 1, 2, . . . ), (36) ⎪ ∂Y (x, 0) 1 ⎪ ⎭ ϕi (x)dσ(x) Bi = pi S ∂t where ∂Y (x, 0) = η1 (x) − pu(x) cos α. (37) ∂t Using the expression (33) for u(x), we can also represent y(x, t) in the form  ∞   ci Ai cos pi t + Bi sin pi t + 2 y(x, t) = sin(pt + α) ϕi (x). (38) pi − p2 i=1 Y (x, 0) = η0 (x) − u(x) sin α,

We can arrive at this expression directly by using the general method discussed in Subsection 3. Indeed, in the case of a pulsating system of forces under consideration we have   fi (t) = ϕi (s)dQ(s, t) = sin(pt + α) ϕi (s) dQ(s) = ci sin(pi t + α) S

S

(i = 1, 2, . . . ), and consequently the general solution Ti (t) (i = 1, 2, . . . ) of (18) has the form Ti (t) = Ai cos pi t + Bi sin pi t +

p2i

ci sin(pt + α) − p2

(39)

(i = 1, 2, . . . ), and since y(x, t) =

∞ 

Ti (t)ϕi (x),

i=1

we again obtain (38). The oscillation described by (27) is called a forced oscillation. This oscillation is not at all identical with the oscillation of a continuum which is originally in equilibrium and is then set in the motion by the pulsating system of forces under consideration. Indeed, as shown by formulas (36) and (37), when η0 (x) ≡ η1 (x) ≡ 0, the coefficients Ai and Bi (i = 1, 2, . . . ) cannot all vanish simultaneously. To explain the term “forced oscillations” it must be noticed that in practice the oscillations occur in the presence of resistance forces. If for example one assumes the resistance

176

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

forces to be proportional to velocity, with a very small proportionality coefficient, it can be shown that in this case the oscillation consists of two components: y(x, t) = u(x) sin(pt + α) + Y (x, t), where u(x) and α differ little from u(x) and α, whereas Y (x, t), which differs little from Y (x, t) during the first instants, will damp out rapidly with time (in fact exponentially). Thus, after the lapse of a certain time interval, there will remain oscillations described by the first term in the right hand side of (35). These oscillations do not depend on the initial conditions, and they are caused and maintained by the pulsating system of external forces, and are therefore naturally called forced oscillations. Let us consider now the particular case in which the continuum is acted upon by only one concentrated pulsating force F sin(pt + α) acting at the point s. Then, by (31), the amplitude deflection u(x) of the forced oscillation will be u(x) = F · Γ(x, s; p2 ). We thus obtain the following mechanical interpretation of the resolvent: The resolvent Γ(x, s; p2 ) gives the deflection amplitude at the point x of the forced oscillation due to a concentrated pulsating force with unit amplitude u and frequency p, applied at the point s. 5. Resonance phenomenon. Let us discuss now the case that the frequency p of the pulsating system of the disturbing forces coincides with one of the natural frequencies pi (i = 1, 2, . . . ). Two cases can occur: 1◦ The differential dQ(s) is not orthogonal to all the amplitude deflections of the natural oscillations with frequency pi = p; i.e., at least one of the numbers  ci = ϕi (s)dQ(s) (pi = p) S

differs from 0. In this case, formula (38) loses its meaning, for in it the terms corresponding to frequencies pi = p become infinite. To obtain the correct formula for the oscillations in this case, we note that the equation T¨i + p2i Ti = ci sin(pt + α), for pi = p, has a solution given not by (39), but by the expression Ti (t) = Ai cos pi (t) + Bi sin pi t −

ci t cos(pt + α). 2pi

Therefore the formula for the oscillations is now written:   ci Ai cos pi t + Bi sin pi t + 2 y(x, t) = sin(pt + α) ϕi (x) pi − p2 pi =p   ci t + Ai cos pi t + Bi sin pi t − cos(pt + α) ϕi (x). 2pi p =p

(40)

i

We see that the deflection will increase indefinitely with time (the phenomenon of resonance).

2. OSCILLATIONS OF A SEGMENTAL CONTINUUM AND OSCILLATORY KERNELS 177

2◦ The differential dQ(s) is orthogonal to all the amplitude deflections of the natural oscillations with frequency pi = p; i.e.,  ϕi (s)dQ(s) = 0 for all pi = p. (41) S

In this case, as follows from (40), the earlier formula (38) remains in force, provided we eliminate from it the terms in which pi = p (ci = 0). In this case there is no resonance; sometimes this case is called quasi-resonance. We notice that if the disturbing forces can be reduced to one concentrated pulsating force, then, as seen from the conditions of quasi-resonance (41), quasiresonance occurs when and only when the point of application of the forces is a node for all natural oscillations with frequency pi = p. 2. Oscillations of a segmental continuum and oscillatory kernels In the case that the continuum is segmental (see Section 6 of Chap. III), the natural oscillations have many remarkable properties. Let us notice first of all the following: I. The frequencies of the natural harmonic oscillations are all different from each other, p0 < p1 < . . . II. The fundamental tone (p = p0 ) has no nodes. III. The j-th overtone (p = pj ) has exactly j nodes, (j = 1, 2, . . . ). IV. In an oscillation obtained by superposition of the natural oscillations with frequencies pk < pl < · · · < pm , the deflection at any instant of time has at least k nodal places and at most m zero places in I, where I is the set of movable points of the continuum). V. The nodes of two neighboring overtones alternate. All these properties, with the exception of V, were already proved in Chap. III with the aid of the theory of oscillatory matrices for the particular case that the entire mass carried by the continuum is concentrated at a finite number of points of this continuum7 . Proceeding to the general case, we recall that the amplitude function ϕ(x) of the natural oscillation with frequency p is a fundamental function of the loaded integral equation  b ϕ(x) = λ K(x, s)ϕ(s)dσ(s) (42) a

that corresponds to the eigenvalue λ = p2 (see Section 1). Here the kernel K(x, s) (a ≤ x, s ≤ b) is the influence function of the continuum, which is oscillatory under normal fastening conditions of the ends of the continuum (see Sections 6-8, Chap. III). In order to formulate a theorem about integral equations (42), from which the oscillatory properties I–V of the continuum follow, we first give a definition of the oscillatory kernel to which we arrive naturally starting with the concept of the oscillatory influence function of the continuum (see definitions 3 and 3 in Sect. 6, Chap. III). 7 Property IV was proved there in a weaker formulation (see Theorem 8 from Sect. 10, Chap. III).

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IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

Here and in what follows we shall relate with each kernel K(x, s) (a ≤ x, s ≤ b) an interval I which contains the following: 1) all interior points of the interval (a, b), and 2) the endpoint a when K(a, a) = 0, and 3) the endpoint b when K(b, b) = 0. In the mechanical interpretation, the interval I is the set of all movable points of the segmental continuum, extending from a to b, for which the kernel K(x, s) (a ≤ x, s ≤ b) is the influence function. Definition 1. The kernel K(x, s) (a ≤ x, s ≤ b) is called oscillatory if the following three conditions are satisfied 1◦ K(x, s) > 0  x 1 x2 ◦ 2 K s1 s2  x 1 x2 3◦ K x1 x2

for x, s ∈ I, {x, s} = {a, b}∗ ,  x < x2 < · · · < xn . . . xn ≥ 0 for a < 1 < b, . . . sn s1 < s2 < · · · < sn  . . . xn > 0 for a < x1 < x2 < · · · < xn < b.8 . . . xn

(43) (44) (45)

Let us analyse conditions 1◦ , 2◦ and 3◦ . From the continuity of the kernel K(x, s) on the closed interval [a, b] it follows directly that (44) remains in force also when a ≤ x1 , s1 and xn , sn ≤ b. A quite nontrivial fact is that 2◦ and 3◦ lead to the possibility of extending inequality 3◦ . That is, we will show that this inequality holds for all points x1 < x2 < · · · < xn in I. The only cases which need additional analysis occur when x1 = a or xn = b, or when we have simultaneously x1 = a and xn = b. Let, for example, x1 = a ∈ I, i.e., K(a, a) = 0. Let us prove that the Fredholm symbols are positive:   a x 2 . . . xn K > 0 (a < x2 < · · · < xn < b). (46) a x2 . . . xn We shall carry out the proof by induction with respect to the order n of the Fredholm symbol. When n = 1 inequality (46) reduces to the inequality K(a, a) > 0, which, according to the condition, is satisfied. Let us assume that inequality (46) is valid for symbols of order n − 1 and that at the same time, in contradiction to what we want to prove, there exist numbers (a 0, K > 0. K c0 c1 . . . cn−2 c2 . . . cn−1 cn Applying Lemma 3 of Sect. 7, Chap. II we conclude that   c1 c2 . . . cn = 0, K c1 c2 . . . cn which is impossible by (45). 8 {x, s} = {a, b} means that x and s cannot simultaneously be the two ends of the interval a and b. In the case of symmetry of the kernel, this can be written: a < x < b, s ∈ I.

2. OSCILLATIONS OF A SEGMENTAL CONTINUUM AND OSCILLATORY KERNELS 179

Thus the inequality (46) is established. We can prove quite analogously the inequalities (46) in the cases that xn = b ∈ I or that both x1 = a ∈ I and xn = b ∈ I. Thus, the inequalities 1◦ , 2◦ and 3◦ lead to the inequalities (x, s ∈ I, {x, s} = {a, b}),  . . . xn ≥0 . . . sn x < x2 < · · · < xn (a ≤ 1 ≤ b; n = 1, 2, . . . ), s1 < s2 < · · · < sn   x1 x2 . . . xn > 0 (x1 , x2 , . . . , xn ∈ I). 3. K x1 x2 . . . xn

1. K(x, s) > 0  x 1 x2 2. K s1 s2

(47) (48)

(49)

Similar inequalities were used to determine the oscillatory character of the influence function in Section 6 of Chap. III. From these inequalities, on the basis of the criterion for a matrix to be oscillatory (Section 7, Chap. II), we obtain the following equivalent definition of the oscillatory kernel (see the analogous Definition 3 in Sect. 7, Chap. 3). Definition 1 . The kernel K(x, s) (a ≤ x, s ≤ b) is called oscillatory if and only if for any collection of x1 , x2 , . . . , xn in I, among which there is at least one interior point9 , the matrix K(xi , xk )n1 is oscillatory. We note furthermore that for an arbitrary oscillatory kernel, the following proposition holds, which we established in Chap. III, p. 156 for an oscillatory influence function: If K(x, s) (a ≤ x, s ≤ b) is an oscillatory kernel, then always     x1 x2 . . . xn x 1 , s 1 < x2 , s 2 < · · · < x n , s n K . (50) >0 s1 s2 . . . sn x1 , . . . , xn , s 1 , . . . , s n ∈ I We can verify the correctness of this statement by repeating word by word the corresponding proof given in the end of Sect. 8, Chap. 3. for oscillatory influence functions. We are now able to formulate the following basic properties of a loaded integral equation of the form  b ϕ(x) = λ K(x, s)ϕ(s)dσ(s) (51) a

with continuous symmetric oscillatory kernel K(x, s) (a ≤ x, s ≤ b) which imply the oscillatory properties I–V (p. 177) of the segmental continuum for an arbitrary (discrete, continuous and combined) distribution of masses: a) The eigenvalues of (51) are all positive and simple: 0 < λ0 < λ1 < λ2 < . . . , b) The fundamental function ϕ0 (x), corresponding to the smallest eigenvalue λ0 , has no zeros in the interval I, c) For every j = 1, 2, . . . , the fundamental function ϕj (x) corresponding to the j-th eigenvalue λj , has exactly j nodes in I and no other zeros in I, 9 This

stipulation is essential only when n = 2.

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IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

k and m (0 ≤ k ≤ m), and for all ck , ck+1 , . . . , d)mFor all integers  m  2  ci > 0 the linear combination ϕ(x) = ci ϕi (x) has at most m zero cm i=k

i=k

places and at least k nodal places in the interval I, e) The nodes of two successive fundamental functions ϕj (x) and ϕj+1 (x) (j = 1, 2, . . . ) alternate. In the next section we shall establish properties a)–e) for the case of a segmental continuum that is loaded everywhere, i.e., for the case that dσ = 0 at all points of the interval (a, b) (i. e., the function σ(s) does not have intervals of constancy in (a, b)10 ). In this case, in proving the properties a)–d) we shall use only conditions 2◦ and 3◦ from among the conditions 1◦ , 2◦ and 3◦ which define the oscillatory kernel (see Definition 1). Under these two conditions, properties a)–e) were established for ordinary integral equations (dσ = ds) by Kellogg [28a,b]. Thus, in the case that dσ = 0 everywhere, the oscillatory properties of the integral equation (51) are established for a formally broader class of kernels than oscillatory kernels, namely for Kellogg kernels. Attention must be called, however, to the fact that this extension of the class of kernels, from oscillatory kernels to Kellogg kernels, may prove to be fictitious since we do not know of a single example of a kernel for which inequalities (44) and (45) are satisfied and inequality (43) is not. In other words, all known Kellogg kernels are oscillatory kernels. Furthermore, it has been established [30g, 31] that under certain general conditions (namely, if a Kellogg kernel is a so-called Green function of some linear differential equation under certain boundary conditions), this kernel must be oscillatory. In the most general case, in which σ(s) is an arbitrary non-decreasing function, the oscillatory properties a)–e) of the integral equation (51) with symmetric oscillatory kernel will be established in Section 4 (Theorem 5). We shall make substantial use of all three defining conditions of the oscillatory kernel (Definition 1). In conclusion, we note that in this book we have confined ourselves to a study of symmetric kernels since only such kernels appear in problems of oscillations of conservative mechanical systems. However, all the oscillatory properties a)–e) hold also for integral equations with continuous non-symmetric oscillatory kernels [15]. This circumstance has made it possible to observe [30k, l] the existence of a broad class of non-selfadjoint boundary value problems of any order n ≥ 2, with spectra and fundamental functions having properties a)–e). All this became possible after the corresponding results were obtained in a simpler case, for selfadjoint boundary value problems [30d-i]. Incidentally, we note that Kellogg himself did not know that the influence functions (the Green functions) of many selfadjoint problems (Kellogg considered only symmetric kernels) belong to the class of kernels that he considered.

3. Oscillatory properties of the vibrations of an everywhere-loaded continuum 1. We first introduce several notions. 10 In this case, in stating property d), it is possible to speak of “zeros” and “nodes” of the function ϕ(x) instead of “zero places” and “nodal places.”

3. VIBRATIONS OF AN EVERYWHERE-LOADED CONTINUUM

181

We recall (see Chap. III, Sect. 5, Definition 2) that continuous functions ϕ0 (x), ϕ1 (x), . . . , ϕn−1 (x), on [a, b] form a Chebyshev system inside (a, b), if for every numbers ci (i = n−1 n−1  2  ci > 0) the function ci ϕi (x) has at most n − 1 zeros 0, 1, . . . , n − 1; i=0

i=0

within the interval (a, b). Closely related to the Chebyshev systems is the notion of a Markov sequence. These sequences of functions played a fundamental role in the investigations of Academician A.A. Markov on the limiting values of integrals [38a, b], in which he established many of their important properties. Definition 2. A sequence of functions (finite or infinite) ϕ0 (x), ϕ1 (x), ϕ2 (x), . . . is a Markov sequence within the interval (a, b) if for every n (n = 1, 2, . . . )11 the functions ϕ0 (x), ϕ1 (x), . . . , ϕn−1 (x) form a Chebyshev system within (a, b). From Lemma 2 of Sect. 5, Chap. III it follows directly that a sequence of functions ϕ0 (x), ϕ1 (x), . . . will be a Markov sequence if and only if all the determinants   ϕ0 ϕ1 . . . ϕn−1 (a < x1 < x2 < · · · < xn < b) (52) Δ x1 x2 . . . xn differ from 0 and are of the same sign εn . 2. To establish the oscillatory properties of the vibrations in the case that the continuum is loaded everywhere (so that any segment of the continuum carries a positive mass), let us prove the following two theorems (compare with [28a,b]): Theorem 1. Let K(x, s) (a ≤ x, s ≤ b) be a symmetric continuous kernel,     x 1 x2 . . . xn x 1 x2 . . . xn K ≥ 0, K >0 s1 s2 . . . sn x1 x2 . . . xn   (53) x1 < x2 < · · · < xn a< < b; n = 1, 2, . . . . s1 < s2 < · · · < sn Suppose that the function σ(s) (a ≤ s ≤ b) is strictly increasing, i.e., dσ = 0 everywhere in (a, b). Then 1◦ All eigenvalues of the integral equation  b K(x, s)ϕ(s)dσ(s) (54) ϕ(x) = λ

and

a

are positive and simple: 0 < λ 0 < λ1 < λ2 < . . . , 2◦ The corresponding fundamental functions ϕ0 (x), ϕ1 (x), ϕ2 (x), . . . form a Markov sequence within (a, b). 11 Not

exceeding the number of functions in the sequence (if this number is finite).

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IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

Theorem 2. If an orthonormal sequence of functions  b  ϕi (x)ϕk (x)dσ(x) = δik ; i, k = 0, 1, . . . ϕ0 (x), ϕ1 (x), ϕ2 (x), . . . a

is a Markov sequence within (a, b) and dσ = 0 everywhere in (a, b), then 1◦ The function ϕ0 (x) has no zeros within (a, b), 2◦ The function ϕj (x) has exactly j nodes in (a, b) and no other zeros within (a, b) (j = 1, 2, . . . ), 3◦ For all integers k and m (0 < k < m) and arbitrary ci (i = k, k + m m   1, . . . , m; c2i > 0) the linear combination ϕ(x) = ci ϕi (x) has at least k i=k

i=k

nodes and at most m zeros in (a, b) (each antinode is counted as two zeros); in particular, if the function ϕ(x) has m different zeros in (a, b), all these zeros are nodes, 4◦ The nodes of any two neighboring functions ϕj (x) and ϕj+1 (x) alternate (i = 1, 2, . . . ). Before proceeding to prove these theorems, we establish several auxiliary propositions, which can be also of independent interest. 3. The following lemma is an analogue of the Perron theorem for the case of a symmetric matrix (see Section 4, Chap. II), and this analogy is retained also in the proof itself. Lemma 1. If the kernel K(x, s) (a ≤ x, s ≤ b) of an integral equation (54) satisfies the inequalities K(x, s) ≥ 0,

K(x, x) > 0

(a < x, s < b),

(55)

then the eigenvalue λ0 of (54) with smallest absolute value is positive, simple, and not equal to the absolute values of the remaining eigenvalues; the fundamental function ϕ0 (x) corresponding to λ0 has no zeros inside (a, b). Proof. We shall use the abbreviated notation:  b b K[ϕ, ϕ] = K(x, s)ϕ(x)ϕ(s)dσ(x)dσ(s), 

a

a b

ϕ(s)ψ(s)dσ(s).

(ϕ, ψ) = a

On the basis of the maximum principle (see Section 1) the integral form K[ϕ, ϕ] on the “sphere” (ϕ, ϕ) = 1 reaches its maximum μ0 > 012 at a fundamental function ϕ0 (x) of the integral equation (54), and to this fundamental function corresponds the smallest positive eigenvalue λ0 = 1/μ0 : 1 = max K[ϕ, ϕ] = K[ϕ0 , ϕ0 ] > 0, λ0 (ϕ,ϕ)=1  b ϕ0 (x) = λ0 K(x, s)ϕ0 (s)dσ(s). μ0 =

a 12 If

ϕ(x) > 0 (a ≤ x ≤ b), then, by (55), K[ϕ, ϕ] > 0; therefore, the maximum μ0 > 0.

(56) (57)

3. VIBRATIONS OF AN EVERYWHERE-LOADED CONTINUUM

183

Let us show now that ϕ0 (x) has no zeros inside (a, b). For this purpose we note that μ0 = K[ϕ0 , ϕ0 ] ≤ K[|ϕ0 |, |ϕ0 |], (58) (|ϕ0 |, |ϕ0 |) = (ϕ0 , ϕ0 ) = 1. From the maximal property of the number μ0 = 1/λ0 it follows that we have equality in (58), and consequently the function |ϕ0 (x)| is also a fundamental function of the integral equation (54) and corresponds to the same eigenvalue λ0 ;  b K(x, s)|ϕ0 (s)|dσ(s). (59) |ϕ0 (x)| = λ0 a

Let us assume now that the function ϕ0 (x) vanishes inside (a, b) and let us denote by x1 (a < x1 < b) such a zero of the function ϕ0 (x), in any neighborhood of which there are such values of x, for which ϕ0 (x) = 0. Then, by the K(x1 , x1 ) > 0 and the continuity of the kernel, we can choose such a value of s1 near x1 , that K(x1 , s1 ) > 0,

ϕ0 (s1 ) = 0.

We arrive at a contradiction, since when x = x1 , the left hand side of (59) vanishes, while the right hand side is positive (the integrand is not negative in (a, b), and is positive when s = s1 ). We have thus proved, that every fundamental function corresponding to the eigenvalue λ0 has no zeros inside (a, b). It follows directly from this proof that the fundamental functions corresponding to the number λ0 retain their sign in (a, b), and therefore there are no two among these functions that are orthogonal to each other, i.e., λ0 is a simple eigenvalue. Furthermore, it follows from the maximum principle that λ0 is less than all the other positive eigenvalues of the integral equation (54). Let us show that λ0 < |λ |,  where λ is any negative eigenvalue. We denote by ψ(x) the normalized fundamental function, corresponding to the eigenvalue λ :  b  K(x, s)ψ(s)dσ(s) [(ψ, ψ) = 1]. (60) ψ(x) = λ a

Hence 



|ψ(x)| ≤ |λ |

b

K(x, s)|ψ(s)|dσ(s).

(61)

a

We note that in (61) the inequality is strict for every value of x (a < x < b), for which ψ(x) = 0. In fact, were it not so, then for this fixed value of x the product K(x, s)ψ(s) would not change sign in (a, b). When s = x this product has the same sign as the number ψ(x), since K(x, x) > 0. But then the integral in the right hand side of (60) would have the same sign as ψ(x). This is impossible by the same (60), since λ < 0. Multiplying both sides of (61) by |ψ(x)|dσ(x) and integrating, we get 1 = (|ψ|, |ψ|) < |λ |K[|ψ, |ψ| ] ≤ |λ |K[ϕ0 , ϕ0 ] = from which follows λ0 < |λ |. This proves the lemma.

|λ | , λ0

184

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

Remark. This lemma can be applied without modification to a more general case, when the arguments x and s vary in an arbitrary n-dimensional simplex, instead of an interval. 4. We now introduce the associated kernels for the kernel K(x, s), by analogy with the associated matrices (see Section 13 of Chap. I). We define the n-th associated kernel Kn (X, S) by means of the equation     x ≤ x2 ≤ · · · ≤ x n x 1 x2 . . . xn Kn (X, S) = K a≤ 1 ≤b . s1 s2 . . . sn s1 ≤ s2 ≤ · · · ≤ sn Each of the points X = (x1 , x2 , . . . , xn ) and S = (s1 , s2 , . . . , sn ) runs through the n-dimensional simplex M n defined by the inequalities a ≤ x1 ≤ x2 ≤ · · · ≤ xn ≤ b. If X = (x1 , x2 , . . . , xn ) is an interior point of the simplex M n , then a < x1 < x2 < · · · < xn < b. Let us note two properties of associated kernels which we shall find useful later. a) If three kernels K(x, s), L(x, s) and N (x, s) (a ≤ x, s ≤ b) are related by  b N (x, s) = K(x, t)L(t, s)dσ(t), a

then N



x 1 . . . xn s1 . . . sn

i.e.,

  b  = ... a

a



t2

K

   t1 . . . t n x 1 . . . xn L dσ(t1 ) . . . dσ(tn ); t1 . . . tn s1 . . . sn

 Nn (X, S) =

Kn (X, T )Ln (T, S)dσ(T ) (dσ(T ) = dσ(t1 ) . . . dσ(tn )).

(62)

(62 )

Mn

In particular, b) If K (q) (x, s) is the q-th iterated kernel for K(x, s):  b  b ··· K(x, t1 )K(t1 , t2 ) . . . K(tn−1 , s)dσ(t1 ) . . . dσ(tn−1 ), K (q) (x, s) = a

a

then [Kn (X, S)](q) = Kn(q) (X, S), i.e., the n-th associated kernel of the q-th iterated kernel is the q-th iterated kernel of the n-th associated one. Statement a) follows from the integral identity for two systems of functions ψ1 (t), . . . , ψn (t) and χ1 (t), . . . , χn (t):      χ1 . . . χn ψ1 . . . ψn Δ dσ(t1 ) . . . dσ(tn ) = Δ t 1 . . . tn t 1 . . . tn Mn      b  1 b χ 1 . . . χn ψ1 . . . ψn Δ dσ(t1 ) . . . dσ(tn ) = = ··· Δ (63) t1 . . . tn t 1 . . . tn n! a a  b n   =  ψi (t)χk (t)dσ(t) , a

1

3. VIBRATIONS OF AN EVERYWHERE-LOADED CONTINUUM

185

if one puts in this identity ψi (t) = K(xi , t),

χi (t) = L(t, si )

(i = 1, 2, . . . , n).

Statement b) follows directly from a). Let us prove now the following lemma [49]: Lemma 2. If ϕ0 (x), ϕ1 (x), ϕ2 (x) . . . is a complete orthonormal system of fundamental functions of an integral equation (55) with a symmetric kernel K(x, s) (a ≤ x, s ≤ b),  b ϕi (x) = λi K(x, s)ϕi (s)dσ(s) (i = 0, 1, 2, . . . ), (64) a

then the functions 

ϕ i1 x1

ϕ i2 x2

... ...

Δi1 i2 ...in (X) = Δ  0 ≤ i1 < i2 < · · · < in ; X = (x1 , x2 , . . . , xn ) ∈ M n

ϕ in xn 

 (65)

form a complete orthonormal system of fundamental functions of the integral equation with associated kernel  Φ(X) = Λ Kn (S, X)Φ(S)dσ(S) (66) Mn (dσ(S) = dσ(s1 )dσ(s2 ) . . . dσ(sn )). With this we have



 ϕ i1 . . . ϕ in = x1 . . . xn       ϕ i1 . . . ϕ in x 1 . . . xn Δ dσ(s1 ) . . . dσ(sn ), = λ i1 . . . λ in · · · K s1 . . . sn s1 . . . sn    Δ

(67)

Mn



or Δi1 i2 ...in (X) = λi1 λi2 . . . λin

Kn (X, S)Δi1 i2 ...in (S)dσ(S).

(67 )

Mn

Thus, all possible products λi1 λi2 . . . λin (0 ≤ i1 < i2 < · · · < in ) of n of the numbers λ0 , λ1 , λ2 , . . . form a complete system of eigenvalues of the integral equation (66). Proof. Let us put in the identity (63) ψk (t) = K(xk , t),

χk (t) = ϕik (t)

(k = 1, 2, . . . , n).

We then obtain directly (67) from (64). With the aid of the same integral identity (63) we can readily verify the orthonormality of the functions (65). It remains to prove the completeness of the system of functions (65). Let us consider first the case that K(x, s) is a positive definite kernel (incidentally, only this case will be encountered from now on).

186

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

We use the Mercer expansion K(x, s) = from which we readily obtain   x 1 x2 . . . xn Kn = s1 s2 . . . sn or Kn (X, S) =

∞  ϕn (x)ϕn (s) , λn n=0

 0≤i1 0

(x, s ∈ Iσ , a < s < b),

(80 )

and the function σ(s) has at least one growth point within (a, b), then the smallest (in absolute value) eigenvalue λ0 of integral equation (76) is positive, simple, and different from the absolute values of all the other eigenvalues. The corresponding fundamental function ϕ0 (x) does not change sign in (a, b) and has no zeros in Iσ . The proof is similar to that of Lemma 4. Corollary. If we replace condition (80 ) of Lemma 4 by the stronger condition K(x, s) > 0 (x ∈ I, s ∈ Iσ , a < s < b), (81)

194

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

then ϕ0 (x) has no zeros in I. Indeed, inserting in (78) an arbitrary x ∈ I, according to (81) we obtain: ϕ0 (x) = 0. 2. For what is to follow we have to generalize somewhat the notions of a Chebyshev system and of a Markov sequence (see Section 5, Chap. II and Section 3, Chap. IV). Definition 3. We shall say that functions ϕ0 (x), ϕ1 (x), . . . , ϕn−1 (x) form a Chebyshev system in the interval I with respect to Iσ , if for every ci (i = n−1 n−1  2  ci > 0) the function ci ϕi (x) changes sign in the inter0, 1, . . . , n − 1; i=0

i=0

val I at most n − 1 times and has at most n − 1 zeros on the set Iσ . Definition 4. A sequence of functions ϕ0 (x), ϕ1 (x), ϕ2 (x), . . . is called a Markov sequence in the interval I with respect to Iσ , if for every n = 1, 2, . . . , the functions ϕ0 (x), ϕ1 (x), . . . , ϕn−1 (x) form a Chebyshev system in I relative to Iσ . To formulate further propositions it will be convenient to introduce the following notation: for every n = 1, 2, . . . , we denote by Iσn the set of all those points X = (x1 , x2 , . . . , xn ) from M n (X ∈ M n , if a ≤ x1 ≤ x2 ≤ · · · ≤ xn ≤ b), all of whose coordinates xi (i = 1, 2, . . . , n) belong to Iσ . The following lemma is a generalization of Lemma 2 of Sect. 5, Chap. III. Lemma 5. Functions ϕ0 (x), ϕ1 (x), . . . , ϕn−1 (x) form a Chebyshev system in I with respect to Iσ , if and only if the determinant   ϕ0 ϕ1 . . . ϕn−1 (82) Δ x1 x2 . . . xn does not change sign in M n and is different from zero on Iσn . Proof. Indeed, if the determinant (82) does not change sign in M n , then according to Lemma 4 (Sect. 5, Chap. III) the function ϕ(x) =

n−1 

ci ϕi (x)

 n−1 

i=0

 c2i

>0

i=0

can change sign in (a, b) at most n − 1 times. On the other hand, ϕ(x) cannot vanish at n points x1 , x2 , . . . , xn from Iσ , since then the corresponding value of the determinant (82) would be equal to zero. Corollary. If functions ϕ0 (x), ϕ1 (x), . . . , ϕn−1 (x) form a Chebyshev system in the interval I with respect to Iσ , then for arbitrary numbers ci (i = 0, 1, . . . , n − n−1  2 ci > 0) and arbitrary points x1 < x2 < · · · < xs from Iσ , the maximal num1; i=0

ber of sign changes of the function ϕ(x) =

n−1  i=0

ci ϕi (x) at the points x1 , x2 , . . . , xs ,

4. VIBRATIONS OF AN ARBITRARILY LOADED CONTINUUM

195

i.e., the maximal number of sign changes in the sequence 14 ϕ(x1 ), ϕ(x2 ), . . . , ϕ(xs ), does not exceed n − 1. We shall formulate the conclusion of this corollary briefly as  follows: the maxin−1 n−1 %   2 ci ϕi (x) ci > 0 on mal number of sign changes of the function ϕ(x) = i=0

+ Iσ does not exceed n − 1 : Sϕσ ≤ n − 1.

i=0

Proof. Let us assume the opposite, i.e., that there exist n + 1 points in Iσ x1 < x2 < · · · < xn+1 , such that for some integer h we have (−1)h+i ϕ(xi ) ≥ 0

(i = 1, 2, . . . , n + 1).

Then we expand the vanishing determinant:   ϕ ϕ0 ϕ1 . . . ϕn−1 = Δ x1 x2 . . . xn xn+1  n+1  ... ϕ0 = (−1)n+i+1 ϕ(xi )Δ x1 . . . xi−1 xi+1 . . . i=1

ϕn−1 xn+1

(83)

 = 0.

From Lemma 5 and (83) it follows that all terms of this sum have the same sign. At the same time, one of the terms is known to be different from 0, since all numbers ϕ(x1 ), ϕ(x2 ), . . . , ϕ(xn ) cannot simultaneously vanish, for then the determinant (82) would vanish at x1 < x2 < · · · < xn from Iσ . We thus arrive at the contradiction. This proves the corollary. 3. We now state one important property of oscillatory kernels which will be used in the future. Let, as before   x 1 x2 . . . xn (q) K s1 s2 . . . sn be the Fredholm symbol for the q-th iterated kernel:  b  b (q) K (x, s) = ··· K(x, t1 )K(t1 , t2 ) . . . K(tq−1 , s) dσ(t1 ) . . . dσ(tq−1 ). a

a

Then the following statement is true: If K(x, s) (a ≤ x, s ≤ b) is an oscillatory kernel, and the function σ(s) has at least n points of growth in I, then for q ≥ n ≥ 2 we have     x 1 x2 . . . xn x1 < x2 < · · · < x n K (q) >0 (84) from Iσ s1 s2 . . . sn s1 < s2 < · · · < sn or, using the associated kernel, for q ≥ n (see Sect. 3, Subsection 4) Kn(q) (X, S) > 0

(X, S ∈ Iσn ).

(84 )

14 The definition of the maximal number of sign changes in a sequence of numbers is given in Sect. 5, Chap. II.

196

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

Proof. Let X = (x1 , x2 , . . . , xn ) and S = (s1 , s2 , . . . , sn ) be two arbitrary points from Iσn . Then (see Section 3, Subsection 4)       x 1 . . . xn x 1 . . . xn (q) = K ··· K s1 . . . sn t1 . . . tn n n  M  M  (85) q−1 times   (q−1)    (q−1)  t 1 . . . tn t1 . . . tn ...K dσ(t1 ) . . . dσ(tn(q−1) ) ×K t1 . . . tn s1 ... sn or   Kn(q) (X, S) = ··· Kn (X, T1 )Kn (T1 , T2 ) . . . Mn Mn    (85 ) q−1 times

. . . Kn (Tq−1 , S)dσ(T1 ) . . . dσ(Tq−1 ), where

(k)

(k)

Tk = (t1 , t2 , . . . , t(k) (k = 1, 2, . . . , q − 1). n ) The integrand in (85) is always non-negative. Let us show that it is possible ∗ to choose T1∗ , T2∗ , . . . , Tq−1 in Iσn such that the integrand is positive whenever T1 = ∗ T1∗ , . . . , Tq−1 = Tq−1 . For this purpose we make use of the fact that for an oscillatory kernel we always have   x 1 x2 . . . xn K > 0, (86) s1 s2 . . . sn if the points x1 < x2 < · · · < xn ; s1 < s2 < · · · < sn belong to I and satisfy the condition x1 , s 1 < x 2 , s 2 < · · · < x n , s n . (87) The two given points X = (x1 , x2 , . . . , xn ) and S = (s1 , s2 , . . . , sn ) are taken arbitrarily in Iσn so that in general they do not satisfy condition (87). Let us show ∗ that it is possible to choose intermediate points T1∗ , . . . , Tq−1 from Iσn so that any two neighboring points in the sequence ∗ X, T1∗ , T2∗ , . . . , Tq−1 , S

(88)

satisfy (87). We place over each of the given numbers xi one of the symbols: plus, minus, or zero, depending on whether si > xi , si < xi , or si = xi (i = 1, 2, . . . , n). The system of numbers x1 , x2 , . . . , xn thus breaks into successive “positive”, “negative”, and “zero” groups. ∗ We can now state the rule of construction of the points T1∗ , T2∗ , . . . , Tq−1 in the following way: the point T1∗ is obtained from X by replacing the last x in each positive group and the first x in each negative group by an s. In general, the point Tk∗ is obtained from X by replacing the last k in each positive group and the first k in each negative group by the corresponding s. Here, if the group contains l < k numbers, then of course one replaces only those l numbers (k = 1, 2, . . . , n). Example: + + + − − 0 X = (x1 x2 x3 x4 x5 x6 ), T1∗ = (x1 x2 s3 s4 x5 s6 ), T2∗ = (x1 s2 s3 s4 s5 s6 ), ∗ ∗ T3 = · · · = Tq−1 = S = (s1 s2 s3 s4 s5 s6 ).

4. VIBRATIONS OF AN ARBITRARILY LOADED CONTINUUM

197

We always have (not only in this example) ∗ ∗ = · · · = Tq−1 = S. Tn∗ = Tn+1 ∗ it follows that From the rule of the construction of the points T1∗ , . . . , Tq−1 every two adjacent points of the sequence (88) satisfy the condition (87), and since ∗ all these points belong to Iσn , it follows that when T1 = T1∗ , . . . , Tq−1 = Tq−1 , the  integrand in (85) assumes a positive value. Inequalities (84) and (84 ) now follow.

4. We are now able to prove the following theorem: Theorem 4. If K(x, s) (a ≤ x, s ≤ b) is a continuous symmetric oscillatory kernel, and the function σ(s) has at least one point of growth inside (a, b) then 1◦ The integral equation (76) has an infinite set of eigenvalues, if the function σ(s) has an infinite set of growth points in I. If Iσ contains a finite number p of points, then the equation (76) has p eigenvalues, 2◦ All the eigenvalues of (76) are positive and simple: 0 < λ 0 < λ1 < λ2 < . . . ,

(89)

◦

3 The corresponding fundamental functions ϕ0 (x), ϕ1 (x), ϕ2 (x), . . . form a Markov sequence in the interval I with respect to Iσ . Proof of 1◦ . If the function σ(s) has a total of p growth points s1 < s2 < · · · < sp in I with jumps mi (i = 1, 2, . . . , p) at these points, then the integral equation (76) is equivalent to the equation ϕ(x) = λ

p 

K(x, sk )mk ϕ(sk )

(a ≤ x ≤ b).

(90)

k=1

By taking here x equal to s1 , s2 , . . . , sp , we obtain a system p of linear algebraic equations:   p  1 (i = 1, 2, . . . , p), (91) μϕ(xi ) = K(si , sk )mk ϕ(sk ) μ= λ k=1

with an oscillatory matrix of coefficients K(si , sk )mk p1 . The quantities that are the reciprocals of the eigenvalues of this oscillatory matrix (this number is equal to p) will be the eigenvalues of the integral equation (76). This integral equations has no other eigenvalues in this case. The values of the fundamental functions at the growth points s1 , s2 , . . . , sp are determined from the system (91), and after that all other values are determined by (90). In the case under consideration, the oscillatory properties of the eigenvalues and fundamental functions can be obtained from both the theory of oscillatory matrices (Chap. II and III) and from the general theory of integral equations with oscillatory kernel (see below). If the function σ(s) has an infinite number of growth points, then the integral equation (76) should have an infinite set of eigenvalues. In the contrary case, it would follow from the finite Mercer series (see Section 1) that all the principal Fredholm symbols of sufficiently large order should vanish at the growth points, which contradicts the oscillatory character of the kernel K(x, s).

198

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

Let us proceed now to the proof of 2◦ and 3◦ . We enumerate the eigenvalues in the order of non-increasing absolute values, repeating each eigenvalue according to its multiplicity: |λ0 | ≤ |λ1 | ≤ |λ2 | ≤ . . . The corresponding fundamental functions will be denoted by ϕ0 (x), ϕ1 (x), ϕ2 (x), . . .

(92)

The oscillatory kernel satisfies the conditions of Lemma 4 and its conclusions. Therefore 0 < λ0 < |λ1 | (93) and ϕ0 (x) does not have any zeros in I. Let us consider first the case that σ(s) has an infinite set of growth points. (q) Then for every n = 2, 3, . . . and for every q ≥ n, the kernel Kn (X, S) (X, S ∈ M n ) satisfies the conditions of Lemma 4, since Kn(q) (X, S) ≥ 0 (X, S ∈ M n ), Kn(q) (X, S) > 0 (X, S ∈ Iσn ). Therefore (see analogous arguments in the proof of Theorem 1) 0 < λq0 λq1 . . . λqn−1 < |λq0 λq1 . . . λqn−2 λqn |

(n = 2, 3, . . . , q ≥ n).

(94)

Taking for q an odd number, we obtain inequalities (89) from (93) and (94). In addition, the function   ϕ0 ϕ1 . . . ϕn−1 Δ0,1,...,n−1 (X) = Δ (X = (x1 , x2 , . . . , xn ) ∈ M n ) (95) x1 x2 . . . xn (q)

is the fundamental function of the kernel Kn (X, S), corresponding to the smallest eigenvalue λ0 λ1 . . . λn−1 . Therefore, by the same Lemma 4, the determinant (95) does not change sign in M n when n ≥ 2, and is different from zero on Iσn . This is also the case that n = 1, if we take Δ = ϕ0 (x). According to Lemma 5, the functions (92) form a Markov sequence in I with respect to Iσ . If the function σ(s) has a finite number p of growth points in I, then all of our arguments are valid as long as n ≤ p; i.e., for all p eigenvalues and corresponding fundamental functions (see 1◦ ). This proves the theorem completely. 5. Let ϕ0 (x), ϕ1 (x), ϕ2 (x), . . .

(96)

be a complete orthonormal system of fundamental functions of the integral equation (76) with an oscillatory kernel K(x, s) (a ≤ x, s ≤ b), corresponding to the eigenvalues (0 0. K s1 s2 . . . sn In particular, inequality (97) holds at all points X = (x1 , s2 , . . . , xn ) (x1 < x2 < · · · < xn from I), for each of which one can find such a point S = (s1 , s2 , . . . , sn ) from Iσn , that (98) x1 , s 1 < x 2 , s 2 < · · · < x n , s n . If K(x, s) is the influence function of a string, then this condition exhausts all the points X = (x1 , x2 , . . . , xn ) ∈ M n for which the inequality (97) is satisfied, for all Fredholm symbols vanish in the case that condition (98) is violated (see Chap. III, p. 155). If K(x, s) is the influence function of a rod, conditions (98) are replaced by the following conditions (see Chap. III, p. 155):  x1 < x 2 < · · · < x n , s 1 < x 2 < · · · < x n , . (99) xk < sk+2 , sk < xk+2 (k = 1, 2, . . . , n − 2) In this case (97) holds if, for the point X = (x1 , x2 , . . . , xn ) (x1 , x2 , . . . , xn

from I)

there exists at least one point S = (s1 , s2 , . . . , sn ) from Iσn such that (99) holds. Let now again K(x, s) be an arbitrary oscillatory kernel. Let us notice the following important property of the determinant:   ϕ0 ϕ1 . . . ϕn−1 Δ = x1 x2 . . . xn   ϕ0 (x2 ) ... ϕ0 (xn )   ϕ0 (x1 ) (100)   ϕ1 (x2 ) ... ϕ1 (xn )   ϕ (x ) = 1 1  (x1 < x2 < · · · < xn and I).  .....................................    ϕn−1 (x1 ) ϕn−1 (x2 ) . . . ϕn−1 (xn ) If for a certain p < n, the principal minors of the determinant (100) satisfy     ϕ0 . . . ϕp ϕ0 . . . ϕp−1  0, Δ = = 0, (101) Δ x1 . . . xp x1 . . . xp+1 then for every number x∗ , satisfying the inequalities xp ≤ x∗ ≤ xp+1 , the sequence of values of the functions ϕi (x) at the point x∗ : ϕ0 (x∗ ), ϕ1 (x∗ ), . . . , ϕn−1 (x∗ ),

200

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

is a linear combination of the first p columns of the determinant (100). In particular, the p + 1-st column of determinant (100) is a linear combination of the preceding p columns. Indeed, it follows from (101), as noted above, that   x1 x2 . . . xp+1 K = 0 (s1 < s2 < · · · < sp+1 from Iσ ). s1 s2 . . . sp+1 Let us consider the matrix   ... K(x1 , sp+1 )   K(x1 , s1 )    ................................    ... K(xp , sp+1 )   K(xp , s1 )   ∗ ∗ ... K(x , sp+1 )   K(x , s1 )   K(xp+1 , s1 ) . . . K(xp+1 , sp+1 )

(s1 < s2 < · · · < sp+1

from Iσ ).

Since the kernel K(x, s) is oscillatory, it follows that this matrix is totally non-negative and that in the last row of this matrix there are non-zero elements. Therefore, applying Lemma 1 of Sect. 6, Chap. II15 , we obtain   x∗ x1 . . . xp =0 (s1 , s2 , . . . , sp+1 ∈ Iσ ). K s1 . . . sp sp+1 But then, from (67), with n = p + 1 and xp+1 replaced by x∗ , we obtain   ϕi1 ϕi2 . . . ϕip+1 =0 (0 ≤ i1 < i2 < · · · < ip+1 ≤ n). Δ x1 x2 . . . x∗

(102)

Inasmuch as the first p columns of the determinant (100) are linearly independent, according to condition (101), our statement follows from (102). 6. For a further investigation of the properties of the fundamental functions we shall find it convenient to use the following generalization of Lemma 3 of Section 5 of Chap. III. Lemma 6. Let



b

Φ(x) =

K(x, s)ϕ(s)dσ(s), a

where ϕ(x) is a continuous function on [a, b] which does not vanish simultaneously at all points of Iσ , and changes sign n−1 times on Iσ , and let K(x, s) (a ≤ x, s ≤ b) be a continuous kernel. Then, if     x1 x 2 . . . x n x1 < x2 < · · · < x n K >0 in I , (103) s1 s2 . . . sn s1 < s2 < · · · < sn the number of zeros of Φ(x) in I does not exceed n − 1. If     x1 < x2 < · · · < xn x1 x2 . . . xn ≥0 a≤ K ≤b s1 s2 . . . sn s1 < s2 < · · · < sn

(104)

with strict inequality for some x1 , . . . , xn and s1 , . . . , sn , then the number of sign changes of Φ(x) in the interval (a, b) does not exceed n − 1; i.e., SΦ ≤ Sϕσ . 15 In Lemma 1 we deal with square matrices. We may, however, apply this lemma also to our matrix by repeating the last column on its right side.

4. VIBRATIONS OF AN ARBITRARILY LOADED CONTINUUM

201

Proof. We choose in Iσ points s1 < s2 < · · · < sn such that ϕ(si )ϕ(si+1 ) < 0 (i = 1, 2, . . . , n − 1). Between si and si+1 there is a nodal place of the function ϕ(x); assume that ξi belongs to this place: si < ξi < si+1 (i = 1, 2, . . . , n − 1). We then divide (a, b) into subintervals (ξi−1 , ξi ) (i = 1, 2, . . . , n; ξ0 = a, ξn = b). Each subinterval contains points from Iσ (in any case, one of the points s1 , . . . , sn ), and the function ϕ(x) does not change sign at the points from Iσ which lie in the interval (ξi−1 , ξi ) (i = 1, 2, . . . , n). The rest of the proof of the first part of the theorem is quite analogous to the proof of Lemma 3, Section 5, Chap. III. We introduce the functions 

ξi

Φi (x) = Φ(x) =

K(x, s)ϕ(s) dσ(s)

(i = 1, 2, . . . , n),

ξi−1 n 

Φi (x)

i=1

and, using the integral identity   Φ 1 Φ 2 . . . Φn Δ = x1 x2 . . . xn   ξn  ξ1  x 1 x2 . . . xn = ϕ(s1 ) . . . ϕ(sn )dσ(s1 ) . . . dσ(sn ), ··· K s1 s2 . . . sn ξn−1 ξ0 conclude on the basis of Lemma 2 of Section 5 of Chap. III that the functions Φ1 (x), Φ2 (x), . . . , Φn (x) form a Chebyshev system in the interval I. Therefore the number of zeros of n  Φ(x) = Φi (x) in the interval I does not exceed n − 1. i=1

The second part of the theorem is obtained from the first by approximating the kernel X(x, s) by means of the kernel 

b



Nt (x, s) =

b

Lt (x, u)K(u, v)Lt (v, s)dσ(u)dσ(v) a

(a ≤ x, s ≤ b),

a

where Lt (x, s) is the totally positive kernel of the heat conduction theory, which we introduced in Sect. 5, Chap. III. If the kernel K(x, s) satisfies the conditions (104) of the second part of the theorem, then inequality (103) is satisfied for the kernel Nt (x, s) since         x 1 x2 . . . xn x 1 . . . xn u 1 . . . un Nt = K × Lt s1 s2 . . . sn u 1 . . . un v1 . . . vn Mn Mn   v1 . . . vn dσ(u1 ) . . . dσ(un )dσ(v1 ) . . . dσ(vn ) ×Lt s1 . . . sn   x < x2 < · · · < x n a≤ 1 ≤b . s1 < s2 < · · · < sn

202

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

Therefore, applying the first part of the theorem to the kernel Nt (x, s) and noticing (see Section 5, Chap. III) that lim Nt (x, s) = K(x, s),

t→+0

we obtain the second part of the theorem. 7. We can use Theorem 4 to prove the following fundamental theorem. Theorem 5. If K(x, s) (a ≤ x, s ≤ b) is a continuous symmetric oscillatory kernel, then the loaded integral equation (76) with an arbitrary non-decreasing function σ(s) which has at least one point of growth within (a, b), has the following properties: 1◦ All eigenvalues of the equation (76) are positive and simple: (0 0; 0 ≤ k ≤ m) the function m 

i=k

ci ϕi (x) has at most m zero places and at least k nodal places in I; if the

i=k

number of zero places is m, then all these places are nodal ones, 5◦ The nodes of the functions ϕj (x) and ϕj+1 (x) alternate (j = 1, 2, . . . ). Proof. Statement 1◦ was proved in Theorem 4. Proposition 2◦ follows directly from Lemma 4 and the corollary of that lemma. Proof of 3◦ . On the basis of Theorem 4, for every j = 1, 2, 3, . . . , the functions ϕ0 (x), ϕ1 (x), . . . , ϕj (x) form a Chebyshev system in I with respect to Iσ . It follows therefore that ϕj (x) has at most j sign changes in (a, b) and at most j zeros on Iσ . We denote by p (p ≤ j) the number of sign changes of the function ϕj (x) on Iσ . Then the equality  b K(x, s)ϕj (s)dσ(s) ϕj (x) = λj a

allows us to employ the second part of Lemma 6 and to conclude that the number of sign changes of the function ϕj (x) in (a, b) is also p. Let us prove that p = j. We choose points s1 < s2 < · · · < sp+1 in Iσ such that ϕj (si )ϕj (sj+1 ) < 0

(i = 1, 2, . . . , p).

In each of the intervals (si , si+1 ) there is one and only one nodal place of the function ϕj (x); we denote by xi (i = 1, 2, . . . , p) a point from this nodal place: s1 < x1 < s2 < x2 < · · · < sp < xp < sp+1 .

(105)

4. VIBRATIONS OF AN ARBITRARILY LOADED CONTINUUM

Let us consider the determinant  ϕ0 . . . Δ x1 . . .

ϕp+1 xp

ϕp x

 =

p 

ci ϕi (x).

203

(106)

i=0

We shall show that this determinant does not vanish for any value x = x∗ ∈ I, other than x1 , x2 , . . . , xp . For this purpose we arrange the numbers x1 , . . . , xp , x∗ in increasing order and we denote them by x1 , x2 , . . . , xp+1 . Then, by (105), x1 , s1 < x2 , s2 < · · · < xp+1 , sp+1 , and since s1 , s2 , . . . , sp+1 ∈ Iσ , we have (see remark on p. 200)   ϕ0 ϕ1 . . . ϕp−1 ϕp = 0. Δ x1 x2 . . . xp x∗

(107)

Thus the function (106) has nodes at the points x1 , x2 , . . . , xp (i.e., at the zeros of the function ϕj (x)), and has no other zeros in the interval I. Therefore the product   ϕ0 . . . ϕp−1 ϕp (108) ϕj (x) Δ x1 . . . xp x does not change sign in (a, b) and does not vanish at those points of Iσ which are different from the zeros of ϕj (x)16 . Consequently, in the product (108) the factors are not mutually orthogonal, which is possible (since p ≤ j) only if p = j. Thus we may rewrite (107) as     ϕ0 ϕ1 . . . ϕj−1 ϕj ϕ0 ϕ1 . . . ϕj−1 Δ = Δ ϕj (x∗ ) = 0. (109) x1 x2 . . . xj x∗ x1 x2 . . . xj Since the last row of the determinant (109) which contains ϕj (x1 ), . . . , ϕj (xj ), ϕj (x∗ ), cannot consist entirely of zeros, we conclude that ϕj (x∗ ) = 0. Here x∗ is an arbitrary point in I, different from x1 , x2 , . . . , xj . We have thus proved that there are no other zeros of ϕj (x) in the interval I, with the exception of x1 , x2 , . . . , xj . This proves that the nodal places of the function ϕj (x) reduce to isolated nodes at the points x1 , x2 , . . . , xj . This proves Proposition 3◦ . m 

Proof of 4◦ . Let ϕ(x) =

ci ϕi (x) (

i=k

m 

c2i > 0). Assume that the function

i=k

ϕ(x) has m + 1 zeros in the interval I: ϕ(xi ) = 0

(i = 1, 2, . . . , m + 1; x1 < x2 < · · · < xm+1 ).

This is possible only when  ϕ0 Δ x1

ϕ1 x2

... ...

ϕm xm+1

 = 0.

16 Such points exist in I , if I contains at least j + 1 points (see Theorem 4, item 1◦ ), since σ σ the number of zeros of ϕj (x) on Iσ does not exceed j.

204

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

When moving from m + 1 to m, from m to m − 1, etc., we find a p ≤ m such that     ϕ0 . . . ϕp−1 ϕ0 . . . ϕp Δ = 0, Δ = 0, x1 . . . xp x1 . . . xp+1 but then, as was established earlier (p. 200), for every x∗ in the interval [xp , xp+1 ] we can find such numbers d1 , d2 , . . . , dp , that ϕi (x∗ ) =

p 

dj ϕi (xj )

(i = 0, 1, . . . , m).

j=1

Multiplying both sides of (110) by ci and summing from k to m, we obtain ϕ(x∗ ) =

p 

dj ϕ(xj ) = 0.

j=1

Thus, every number x∗ of the interval [xp , xp+1 ] is a zero of the function ϕ(x); i.e., the numbers xp and xp+1 belong to the same zero place of the function ϕ(x). From the fact that m + 1 zeros of the function ϕ(x) cannot belong to m + 1 different zero places, it follows that the function ϕ(x) has at most m zero places. Let us consider the case that the function ϕ(x) has a maximal number m of zero places in I. Let x1 < x2 < · · · < xm from I belong to different zero places of ϕ(x). Then, on the basis of our preceding arguments,   ϕ0 ϕ1 . . . ϕm−1 = 0. (110) Δ x1 x2 . . . xm In this case the system of linear equations which is satisfied by the coefficients c0 = 0, . . . , ck−1 = 0, ck , ck+1 , . . . , cm of the function ϕ(x) =

m 

ci ϕi (x)

i=0

is

m 

ci ϕi (xj ) = 0

(j = 1, 2, . . . , m).

i=0

This system has rank m and therefore defines the linear combination ϕ uniquely up to a constant multiple, and so   ϕ0 ϕ1 . . . ϕm−1 ϕm . ϕ(x) ≡ cΔ x1 x2 . . . xm x It is seen from this representation that all the zero places of the function ϕ(x) located in the interval (a, b) are nodal places. In order to conclude the proof of 4◦ it remains to show that the function m   2 ϕ(x) = ci ϕi (x) ( m i=k ci > 0) always has at least k nodal places. Without loss i=k

of generality, we may assume that ck = 0. We introduce the notation  b ψ1 (x) = K(x, s)ϕ(s) dσ(s) a

5. HARMONIC OSCILLATIONS OF MULTIPLY SUPPORTED RODS

and, in general,  ψn (x) =

205

b

K(x, s)ψn−1 (s)dσ(s) (n = 1, 2, . . . ; ψ0 (x) ≡ ϕ(x)).

(111)

a

Then

ck cm ϕk (x) + · · · + n ϕm (x). n λk λm

(112)

Sϕ ≥ Sϕσ ≥ Sψ1 ≥ Sψ1 σ ≥ Sψ2 ≥ . . . .

(113)

ψn (x) = According to Lemma 6,

But lim λnk ψn (x) = ck ϕk (x),

n→∞

And since 2◦ implies that the limit function ck ϕk (x) has exactly k nodes, we see that for all sufficiently large n, Sψn ≥ k, and, consequently, by (113), Sϕ ≥ k. This proves Proposition 4◦ . Proof of 5◦ . From 3◦ and 4◦ it follows that for every j = 1, 2, . . . , the functions ϕj (x) and ϕj+1 (x) satisfy the conditions of Lemma 3 of Section 3. Therefore, zeros of these functions alternate. This completely proves the theorem. Remark. As a by-product, we have established (see (109)) that   ϕ0 ϕ1 . . . ϕj−1 Δ = 0, x1 x2 . . . xj if x1 < x2 < · · · < xj are nodes of the function ϕj (x). This implies the existence of such a point S = (s1 , s2 , . . . , sj ) ∈ Iσj , that   x 1 x2 . . . xn = 0, K s1 s2 . . . sn and, consequently, in the case of a rod, xk < sk+2 ; sk < xk+2

(k = 1, 2, . . . , j − 2),

and in the case of a string x1 , s 1 < x 2 , s 2 < · · · < x j , s j . 5. Harmonic oscillations of multiply supported rods 1. If at n arbitrary movable points c1 < c2 < · · · < cn of the continuum S, we introduce stationary (hinged) supports, then the influence function K ∗ (x, s) of the continuum S ∗ with these additional constraints is expressed in terms of the influence function K(x, s) of the continuum S by (56) of Section 3, Chap. III; i.e.,   x c1 c2 . . . cn K s c1 c2 . . . cn  .  K ∗ (x, s) = c 1 c2 . . . cn K c1 c2 . . . cn

206

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

The squares of the natural frequencies λ∗j = (p∗j )2 (j = 0, 1, 2, . . . ) and the corresponding amplitude functions ϕ∗j (x) (j = 0, 1, 2, . . . ) of the continuum S ∗ are obtained as the eigenvalues and the corresponding fundamental functions of the integral equation  l ϕ(x) = λ K ∗ (x, s)ϕ(s) dσ(s), (114) 0

where dσ(s) is the mass of an element ds. If it is assumed that S is a rod (and not a string), which is so fastened that it has properties A and B (Section 6 of Chap. III) and consequently has an oscillatory influence function, then S ∗ will be an n-span beam, for which one can establish several oscillation theorems, analogous to those theorems for the segmental continuum which has been treated in the preceding sections. For this purpose we can use the device indicated in Section 11 of Chap. III, based on the use of a discontinuous function ε(x), which has alternate values ±1 in successive spans of the multiply supported rod S ∗ . Putting ˜ ϕ(x) ˜ = ε(x)ϕ(x), K(x, s) = ε(x)ε(s)K(x, s), we can rewrite (114) in the form 

l

˜ K(x, s)ϕ(s)dσ(s). ˜

ϕ(x) ˜ =λ

(115)

0

As K ∗ (cj , x) = K ∗ (x, cj ) ≡ 0 (0 ≤ x ≤ l; j = 1, 2, . . . , n), ˜ the kernel K(x, s) (0 ≤ x, s ≤ l) is everywhere continuous. By extending naturally the notion of an oscillatory kernel introduced in Section ˜ 6 of Chap. III and in this chapter (Section 2), we can state that the kernel K(x, s) is also oscillatory, namely in the following sense: ˜ 1◦ K(x, s) > 0  ˜ x 1 x2 2◦ K x1 x2  ˜ x 1 x2 3◦ K s1 s2

for ... ... ... ...

x, s ∈ I,  xn > 0 for x1 < x2 < · · · < xn ; xi ∈ I (i = 1, 2, . . . , n), xn  x < x2 < · · · < x n xn ≥ 0 for 0 ≤ 1 ≤ l, sn s1 < s2 < · · · < sn

where I is the set of all movable points of the rod S ∗ . Properties 1◦ , 2◦ and 3◦ hold because for all numbers x1 < x2 < · · · < xn from ˜ i , xk )m is oscillatory (see Section 11, Chap. III). I the matrix K(x 1 It is easy to see that the results of the preceding section concerning integral equations with oscillatory kernels can be extended to equations (115), provided they are suitably reformulated (see also Section 11 of Chap. III on this subject). We thus obtain Theorem 6. For each arrangement of the supports the successive frequencies p∗j (j = 0, 1, 2, . . . ) of the natural oscillations of the multiply supported beam S ∗ are all distinct. The amplitude functions ϕj (x) (j = 0, 1, 2, . . . ) corresponding to these oscillations have the following properties:

5. HARMONIC OSCILLATIONS OF MULTIPLY SUPPORTED RODS

207

1◦ The function ϕ˜j (x) = ε(x)ϕ∗j (x) (j = 0, 1, 2, . . . ) has exactly j nodes 17 and no other zeros except those at the stationary points,18 2◦ The nodes of two successive functions ϕ˜j (x) and ϕ˜j+1 (x) (j = 0, 1, 2, . . . ) alternate. Of course, there is also a property corresponding to property 4◦ of Theorem 5 of the preceding section. We shall not formulate it since it will not be used here. 2. Henceforth we shall denote by S either a rod supported only at the ends with an oscillatory influence function, or a rod obtained from it by introducing a sequence of hinged stationary supports at the points b1 < b2 < · · · < bm . Theorem 7. Let p∗j = p∗j (c) (j = 0, 1, 2, . . . ) be the successive frequencies of the rod S ∗ obtained from the rod S by introducing an additional support at a movable point c (K(c, c) > 0). Then pj < p∗j ≤ pj+1

(j = 0, 1, 2, . . . ),

(116)

where pj (j = 0, 1, 2, . . . ) are the successive frequencies of the rod S. The equality p∗j = pj+1 for a given j occurs if and only if c coincides with a node of the harmonic oscillation of the rod S of frequency pj+1 . Proof. By Sylvester’s identity (see Section 2, Chap. I), the relation    K(x, s) K(x, c)  1 ∗   K (x, s) = K(c, c)  K(c, s) K(c, c)  between the influence functions K(x, s) and K ∗ (x, s) of the rods S and S ∗ implies the relations19     1 x 1 x2 . . . xn c x 1 x2 . . . xn = . (117) K∗ K s1 s2 . . . sn c s1 s2 . . . sn K(c, c) From this we obtain for the Fredholm determinant   l  l ∞  (−λ)n s1 s2 . . . D∗ (λ) = 1 + ··· K∗ s1 s2 . . . n 0 0 n=1

sn sn

 dσ(s1 )dσ(s2 ) . . . dσ(sn )

of the kernel K ∗ (x, s) the expression K(c, c)D∗ (λ) = K(c, c)   l  l ∞  (−λ)n c ··· K + c n! 0 0 n=1

s1 s1

s2 s2

... ...

sn sn

 dσ(s1 )dσ(s2 ) . . . dσ(sn )

= D(c, c; λ). Therefore

17 Some

∞  ϕ2j (c) D∗ (λ) = Γ(c, c; λ) = , K(c, c) D(λ) λj − λ j=0

(118)

of these may be due to the supports c1 , c2 , . . . , cn ; such a case is possible. at the supports c1 , c2 , . . . , cn and at the stationary endpoints. 19 See Sect. 4, Chap. III for the derivation of relations of the type (117) for a more general relation between K ∗ (x, s) and K(x, s). 18 i.e.,

208

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

where ϕj (x) (j = 0, 1, 2, . . . ) is the orthonormal sequence of amplitude functions of the rod S. We assume first that all ϕj (c) = 0

(j = 0, 1, 2, . . . ).

(119)

Then, from (118), it follows that Γ(c, c; λj − 0) = +∞,

Γ(c, c; λj + 0) = −∞

(j = 0, 1, 2, . . . ).

Since, furthermore, we always have ∞  ϕ2j (c) d Γ(c, c; λ) = >0 dλ (λj − λ)2 j=0

and Γ(c, c; λ) > 0 for − ∞ < λ < λ0 , we conclude that inside each of the intervals (λj , λj+1 )

(j = 0, 1, 2, . . . )

∗

the determinant D (λ) has one and only one zero λ∗j (c) = p∗2 j (c)

(j = 0, 1, 2, . . . )

and no other zeros. Thus, under the assumption (119) we have: λj < λ∗j (c) < λj+1

(j = 0, 1, 2, . . . ).

(120)

Let us consider now the case that ϕj (c) = 0 for some j. According to (118), ϕj (c) = 0 if and only if D∗ (λj ) = 0.20 ∗ ∞ If in the sequences of zeros {λj }∞ 0 and {λj }0 we delete the common elements, then the remaining “sifted” sequences will alternate in accordance with the law (120). Hence λj ≤ λ∗j (c) ≤ λj+1 (j = 0, 1, 2, . . . ). It remains to show that equality is excluded in the first inequality. For this we notice that if λj = λ∗j (c) or λj = λ∗j−1 (c), then  l ϕj (c) = λj K(c, s)ϕj (s) dσ(s) = 0, 0

and then ϕj (x) is an amplitude function not only of the rod S, but also of the rod S ∗:  l ϕj (x) = λj K ∗ (x, s)ϕj (s) dσ(s). 0

According to the preceding theorem, the function ϕj (x), according to its position j in the sequence of amplitude functions of the rod S or of the rod S ∗ , should have a specific behavior in the sense of sign alternations. It therefore cannot have the same subscript j with respect to S and S ∗ ; i.e., the equality λj = λ∗j (c) is excluded. ∗

(λ) one takes into account that for λ = λj we have in this case ddtDD(λ) > 0, we can state ∗ that the multiplicity of the zero λj with respect to D (λ) either coincides with the multiplicity of λj with respect to D(λ) or exceeds it by 1. 20 If

5. HARMONIC OSCILLATIONS OF MULTIPLY SUPPORTED RODS

209

This proves the theorem. Remark. From the foregoing arguments we can readily conclude that Theorem 7 in the weakened form, where instead of (116) only pj ≤ p∗j ≤ pj+1

(j = 0, 1, 2, . . . )

(121)

is asserted, holds not only for a rod S with oscillatory influence function, but also for any elastic continuum S. If, in addition, all the frequencies of this arbitrary continuum are simple: p0 < p1 < p2 < . . . , and the point c does not coincide with any of the nodes of its harmonic oscillations, we also have pj < p∗j < pj+1

(j = 0, 1, 2, . . . ).

(122)

Inequalities (121) also follow directly from the minimum properties of the frequencies of the continuum. 3. We obtain from Theorem 7 the following characteristic property of the nodes of the natural oscillations of a rod S. Theorem 8. If n hinged supports are introduced at movable points c1 < c2 < · · · < cn of the rod S, its fundamental tone (j-th overtone) increases, but does not exceed the n-th overtone (or respectively the (n + j)-th overtone) which was present before the introduction of the supports. The fundamental tone (the j-th overtone) after introduction of the supports, reaches the value of that n-th (respectively n + jth) overtone if and only if the supports are placed at the nodes of the latter. Proof. We denote by Sk (k = 1, 2, . . . , n) the rod obtained from S by in(k) troducing hinged supports at the points c1 , c2 , . . . , ck ; we denote by pj (j = (k)

0, 1, 2, . . . ) the successive frequencies of the rod Sk . We also denote by ϕj (x) (j = 0, 1, 2, . . . ) the corresponding orthonormal amplitude functions of this rod. According to the preceding theorem, (k−1)

pj

(k)

< pj

(k−1)

(0)

≤ pj+1

Hence

(j = 0, 1, 2, . . . ; k = 1, 2, . . . , n; pj (n)

pj < pj For any j the equality

≤ pj+n (n)

pj

= pj ).

(j = 0, 1, 2, . . . ). = pj+n

(123)

will hold if and only if (n)

pj

(n−1)

= pj+1

(1)

= · · · = pj+n−1 = pj+n .

(1)

But the equality pj+n = pj+n−1 means (see end of the proof of Theorem 7) that the support c1 is located at a node of the amplitude function ϕj+n (x) and, (1) consequently, ϕj+n (x) = ±ϕj+n−1 (x). (2)

(1)

Analogously, the equality pj+n−2 = pj+n−1 means that the support c2 is located (1)

(1)

(2)

at a node of the function ϕj+n−1 (x) and, consequently, ϕj+n−1 (x) = ±ϕj+n−2 (x). Continuing this argument, we arrive at the conclusion that all the supports c1 , c2 , . . . , cn are located at the nodes of the functions (1)

(n−1)

ϕj+n (x) = ±ϕj+n−1 (x) = · · · = ±ϕj+1 (x).

210

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

This proves the theorem. Remark. If we use the rule indicated in Subsection 1 to set up the function ϕ˜n (x) = ε(x)ϕn (x), it will have n nodes α1 < α2 < · · · < αn inside the interval (0, l). If none of these nodes coincides with any of the points b1 < b2 < · · · < bm of the hinged supports of the rod S, then the points α1 < α2 < · · · < αn will also be nodes of the amplitude function ϕn (x). In this case the equality (n)

p0

= pn

(124)

will take place in accordance with the proved theorem in the case that ci = αi (i = 1, 2, . . . , n). But if at least one of the nodes ϕi coincides with one of the support points b1 < b2 < · · · < bn , then the equality (124) will not hold for any placement of additional supports at the movable points of the rod S. However, if one agrees to understand the placement of the additional support at an already existing hinged support bi to mean a rigid clamping of the rod at the point bi , then we can state again as before that equality (124) occurs if and only if ci = αi (i = 1, 2, . . . , n). We shall not develop this further here. An analogous remark can be made also with respect to the general equality (123), which is thus always attained for certain placements of n additional supports at movable or non-movable interior zeros of the amplitude function ϕn+j (x), and (n + j)! such placements. there will be exactly n!j! 6. Oscillatory properties of forced vibrations 1. Let S be a segmental continuum with an oscillatory influence function. We then have the following: Theorem 9. If at least one end of the continuum S is movable, then for every harmonic oscillation of S, the amplitude of the oscillations of this end is non-zero. Furthermore, for any free oscillation of the continuum S this end cannot remain at rest for all time. Proof. Let us assume that the end x = 0 is movable (K(0, 0) > 0). Then, by Theorem 5, all the amplitude functions of the rod S are different from 0 at the point x = 0, i.e., ϕj (0) = 0 (j = 0, 1, 2, . . . ), which gives the first statement of Theorem 9. According to Section 1, in the case of a free oscillation, the deflection y = y(x, t) is given by: ∞  y(x; t) = Cj ϕj (x) sin(pj t + αj ), j=0

where the series on the right converges absolutely and uniformly in the band 0 ≤ x ≤ l, 0 ≤ t < ∞. In particular, y(0; t) =

∞  j=0

Cj ϕj (0) sin(pj t + αj )

(0 ≤ t < ∞),

6. OSCILLATORY PROPERTIES OF FORCED VIBRATIONS

211

so that

 2 T y(0, t) sin(pj t + αj ) dt. T →∞ T 0 If y(0; t) ≡ 0 (0 ≤ t ≤ ∞), then all Cj = 0 (j = 0, 1, 2, . . . ) and consequently, y(x, t) ≡ 0 (0 ≤ x ≤ l; 0 < t < ∞). This proves the theorem. Cj ϕj (0) = lim

Using stronger analytical means (in particular, the results of the article by B. Ya. Levin [36]) we can show the following: a) In the case of a rod, a movable end cannot remain at rest for any interval of time, no matter how short, b) In the case of a string (which is loaded arbitrarily) a movable end cannot remain at rest during a time interval greater than a certain quantity L. If the quantity L is chosen such as to have the minimum possible value21 , it is found to be independent of the conditions of fastening of the string at the ends. The quantity l/L can be called the average velocity of propagation of disturbances over the string (for a specified tension and distribution of masses). Similarly, statement a) can be interpreted as the assertion that under the assumed idealization of the phenomenon of transverse oscillation of the rod, transverse disturbances are transmitted instantaneously along the rod. The proof of Propositions a) and b) would lead us too far afield and we shall omit it. 2. If we introduce at a movable end of the continuum S a hinged support, then by Theorem 7 (see also the remark to this theorem), the frequencies p∗j (j = 0, 1, 2, . . . ) obtained will alternate with the previous frequencies of S: p0 < p∗0 < p1 < p∗1 < p2 < . . . The principal task of this section is to establish the following Theorem 10. The forced oscillation which results when a concentrated force, pulsating harmonically with a non-resonant frequency p, is applied to a movable end of the continuum S, has exactly j nodes where j is the subscript of the first frequency p∗j greater than p, and has no other stationary points in I.22 Furthermore, a) If p∗j−1 < p < pj , then the phase of the oscillation of the movable end coincides with the phase of the disturbing force; b) if pj < p < p∗j , then the indicated phases are directly in opposition, and finally, c) if for some j p = p∗j , then the end under consideration is stationary. Statement c) appears at first glance quite paradoxical: the point of application of the disturbing force remains at rest, while the other points are in motion! L > 0 if and only if σ  (t) > 0 on a set of positive measure. is the set of movable points of the continuum S in the sense of the definition given in Section 2. 21 Here 22 I

212

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

However, this paradox disappears if it is recalled, first, that the forced oscillation is the oscillation which remains after the damping of the free oscillations (in the presence of very small resistance forces) and, second, that p∗j is the frequency of the j-th natural oscillation of the continuum when a hinged support is introduced, consequently, when p = p∗j , the pulsating force, so to speak, replaces the reaction of the absent support. The theorem may be generalized in a natural way to the case of a multiplysupported rod. This generalization is obtained with the aid of a device based on the use of the function ε(x) which was introduced in the preceding section; we shall therefore confine ourselves to the formulation and proof of theorem 10 only for the case of a segmental continuum. As was shown on p. 176, the amplitude deflection u(x) of a forced oscillation under the influence of a pulsating force F sin(pt + α) concentrated at the point s is given by u(x) = F · Γ(x, s; p2 ), where Γ(x, s; λ) is the resolvent of the integral equation of the vibrations of the continuum S. Thus, Theorem 10 states a property of the resolvent Γ(x, s; λ) for the case that the argument s coincides with the movable end of the continuum. To prove Theorem 10 we shall show that this property is possessed at s = c (where c = a or b and K(c, c) > 0) by the resolvent Γ(x, s; λ) of any loaded integral equation23  b ϕ(x) = λ K(x, s)ϕ(s) dσ(s) (125) a

with an oscillatory kernel K(x, s). The role of the numbers λ∗j = p∗2 j (j = 0, 1, 2, . . . ) in such a general setting will be played by the successive roots of the equation: Γ(c, c; λ) =

∞  ϕ2j (c) = 0, λj − λ j=0

(126)

or, what is the same (see Subsection 2 of the preceding section), the eigenvalues of the integral equation:  b ϕ(x) = λ K ∗ (x, s)ϕ(s)dσ(s), a

   K(x, s) K(x, c)  1 .  K (x, s) = K(c, c)  K(c, s) K(c, c)  Thus, in the general theory of loaded integral equations (125) with oscillatory kernel K(x, s), the main statement of the oscillation Theorem 10 corresponds to the following: where

∗

Theorem 10 . The resolvent Γ(x, s; λ) of a loaded integral equation (125) with oscillatory kernel K(x, s), which is different from zero at the point x = s = c (where c = a or b) has the property: 23 It

is assumed that σ(s) has at least one growth point inside (a, b).

6. OSCILLATORY PROPERTIES OF FORCED VIBRATIONS

213

For each λ = λj from the interval λ∗j−1 < λ < λ∗j

(j = 0, 1, 2, . . . )

the function Γ(x, c; λ) of the variable x has exactly j nodes between a and b and has no other zeros in I. Since λ∗j (j = 0, 1, 2, . . . ) are simple roots of (126), − ∞ < λ < λ0 ,

Γ(c, c; λ) > 0 for and Γ(c, c; λj − 0) = +∞, it follows that Γ(c, c; λ)



Γ(c, c; λj + 0) = −∞,

> 0 for λ∗j−1 < λ < λj , < 0 for λj < λ < λ∗j

(j = 0, 1, 2, . . . ; λ∗−1 = 0).

(127)

For the case that K(x, s) is the influence function of a segmental continuum, and σ(x) specifies the distribution of masses on it, inequality (127) yields statements a), b), and c) of Theorem 10. The proof of the general Theorem 10 requires relatively extensive considerations. These considerations have as their origin certain simple mechanical ideas. The mechanical pattern of the proof will become clearer if we make the following preliminary remark. If K(x, s) is the influence function of the continuum S, and x = c is a point on it, then the kernel    K(x, s) 1 K(x, c)   LA (x, s) = (A > 0) (128) A + K(c, c)  K(c, s) K(c, c) + A  is the influence function of a continuum SA , obtained from S by introducing at the point x = c, an elastic support (spring) whose reaction R is proportional to the deflection at the point x = c: 1 R= y(c). (129) A Indeed, the influence function L(x, s) of the continuum SA specifies the deflection at the point x of the continuum S under the influence of a unit force applied at the point s and the reaction R = R(s) at the point x = c; consequently L(x, s) = K(x, s) − RK(x, c).

(130)

On the basis of (129) we have L(c, s) = AR,

that is K(c, s) − RK(c, c) = AR.

Determining R from this equation and substituting its value into (130), we find that L(x, s) coincides with LA (x, s). We notice that when A = 0, the kernel LA (x, s), as expected, becomes the known influence function of the continuum S ∗ , which is obtained from S by introducing a stationary support at the point x = c. If K(x, s) is an oscillatory influence function, and the point c coincides with one of the ends of S, then it is natural to expect that LA (x, s) will be an oscillatory function. Indeed, we have the following general proposition:

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IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

Lemma 7. If K(x, s) (a ≤ x, s ≤ b) is an oscillatory kernel and K(c, c) > 0 (c = a or b), then the kernel LA (x, s) (A ≥ 0) determined by (128) is also oscillatory. Proof. If in the Sylvester identity:      a00 a01 . . . a0n   b11 b12 . . . b1n     1  a10 a11 . . . a1n   b21 b22 . . . b2n   = , a00  . . . . . . . . . . . . . . . . . . .   . . . . . . . . . . . . . . . . . . .      an0 an1 . . . ann bn1 bn2 . . . bnn where bik

1 = a00

  a00   ai0

 a0k  aik 

(i, k = 1, 2, . . . , n),

we put a00 = K(c, c) + A,

we then obtain  x 1 x2 . . . LA s1 s2 . . .

xn sn

ai0 = K(xi , c), a0k = K(c, sk ), (i, k = 1, 2, . . . , n),

aik = K(xi , sk )



  K(c, sn )   K(c, c) + A K(c, s1 ) . . .   1 K(x1 , s1 ) . . . K(x1 , sn )   K(x1 , c) =  ; K(c, c) + A  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    K(xn , c) K(xn , s1 ) . . . K(xn , sn )

i.e.  LA

x1 s1

 . . . xn . . . sn     x 1 x2 . . . xn c x 1 . . . xn K K c s1 . . . sn s1 s2 . . . sn +A . = A + K(c, c) A + K(c, c) x2 s2

In particular, as follows directly from (128),   c x K c s K(x, s) LA (x, s) = +A . A + K(c, c) A + K(c, c)

(131)

(131 )

It is clear from this that when A > 0, the movable end of the interval (a, b) for the kernel K(x, s) will also be movable for LA (x, s). To the contrary, if A = 0, this holds only for the end that differs from c; for the end c, it will be stationary for L0 (x, s) (L0 (c, c) = 0), although it was movable for K(x, s) (K(c, c) > 0). Taking this into account, we can readily verify using (131) and (50) that the kernel LA (x, s) is oscillatory for every A ≥ 0. After all these preliminary considerations, let us give the

6. OSCILLATORY PROPERTIES OF FORCED VIBRATIONS

215

Proof of Theorem 10 . In parallel with the integral equation (125) let us consider the integral equation:  b ϕ(x) = λ LA (x, s)ϕ(s) dσ(s). (132) a

Substituting here (128) instead of LA (x, s), we find that this equation is equivalent to the system: ⎫  b ⎪ ⎪ ⎪ ϕ(x) − λ K(x, s)ϕ(s) dσ(s) = −ξK(x, c), ⎬ a . (133)  b ⎪ ⎪ ⎪ K(c, s)ϕ(s) dσ(s)⎭ [A + K(c, c)]ξ = λ a

This system has no non-trivial solutions {ϕ(x), ξ} with ξ = 0. Indeed, if ξ = 0 and ϕ(x) ≡ 0, we conclude from the first equation that λ = λj and ϕ(x) = Cϕj (x) (C = 0); but then the second equation of (133) yields ϕj (c) = 0, which is impossible by Theorem 5. Let us notice now that when λ is one of the eigenvalues of (125), the system (133) has only trivial solution ϕ ≡ 0, ξ = 0. Indeed, if λ = λj , then multiplying the first equation by ϕj (s)dσ(s) and integrating over (a, b) we find that −ξφj (c)/λj = 0, i.e., ξ = 0. Therefore, in finding non-trivial solutions {ϕ(x), ξ} of the system (133), we can assume that ξ = 0 and λ = λj (j = 0, 1, 2, . . . ). But in this case the first of the equations can be solved with respect to ϕ for every ξ (in fact, for every right hand side). Solving this by the well-known rules, we find

  b ϕ(x) = −ξ K(x, c) + λ Γ(x, s; λ)K(s, c)dσ(s) = −ξΓ(x, c; λ). (134) a

Introducing this expression for ϕ(x) into the second equation (133) and dividing through by ξ, we obtain the following equation for the eigenvalues λ of (132):  b A + K(c, c) = −λ K(c, s)Γ(s, c; λ) dσ(s), a

or A + Γ(c, c; λ) = 0, in yet another form24 A+

∞  ϕ2j (c) = 0. λj − λ j=0

(135)

24 (135) can be obtained in still another way, namely by computing directly from (131) the Fredholm determinants DA (λ) of (132), similar to what was done in the preceding section for the case A = 0; we then obtain

[A + K(c, c)]DA (λ) = D(c, c; λ) + AD(λ). Thus, the left hand side of (135) is nothing but [A + K(c, c)]

DA (λ) . D(λ)

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IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

Let us denote the left hand side of (135) by Φ(λ). By Theorem 5, ϕ2j (c) > 0

(j = 0, 1, 2, . . . ).

Therefore Φ(λj − 0) = +∞,

Φ(λj + 0) = −∞

(j = 0, 1, 2, . . . ).

In addition, ∞ dΦ  ϕ2j (c) = >0 dλ (λj − λ)2 j=0

(−∞ < λ < ∞).

Thus, in the interval (−∞, λ0 ), the function Φ(λ) increases from A ≥ 0 to +∞ and consequently has no zeros in this interval. In each of the intervals (λj , λj+1 ) (j = 0, 1, 2, . . . ) the function Φ(λ) increases from −∞ to +∞ and consequently has one and only one zero λj (A) (j = 0, 1, 2, . . . ), so that λ0 < λ0 (A) < λ1 < λ1 (A) < λ2 < . . . . All the λj (A) are decreasing functions25 of A, since it follows from (135) that dλ 1 =−  < 0. dA Φ (λ) According to the definition of the numbers λ∗j (j = 0, 1, 2, . . . ), λj (0) = λ∗j

(j = 0, 1, 2, . . . ).

Thus, in view of the monotonicity of the function λj (A) (A ≥ 0) we have λ0 < λ0 (A) ≤ λ∗0 < λ1 < λ1 (A) ≤ λ∗1 < . . . . It is also clear from (135) that lim λj (A) = λj

A→∞

(j = 0, 1, 2, . . . ).

Thus, each λj (A) decreases from λ∗j to λj as A increases from 0 to ∞. Inserting the number λj (A) in place of λ into (134), we obtain the fundamental (A) function ϕj (x) of (132), corresponding to this eigenvalue: (A)

ϕj (x) = const. Γ(x, c; λj (A))

(j = 0, 1, 2, . . . ). (A)

On the other hand, since LA (x, s) is an oscillatory kernel, the function ϕj (x) has exactly j nodes between a and b. By letting A vary from 0 to ∞, we see that for each λ in the interval λj < λ ≤ λ∗j

(j = 0, 1, 2, . . . )

the resolvent Γ(x, c; λ), as a function of x, has exactly j nodes between a and b and has no other zeros in I. To complete the proof of Theorem 10 , it remains to investigate the behavior of Γ(x, c; λ), as a function of x, for values of λ from the interval λ∗j−1 < λ < λj 25 As

(j = 0, 1, 2, . . . ; λ∗−1 = 0).

A increases, the stiffness of the elastic support at the point c decreases (see (129)).

6. OSCILLATORY PROPERTIES OF FORCED VIBRATIONS

217

For this purpose we impose a “supplementary load” on the integral equation (125) at the point x = c by means of a certain mass M > 0; i.e., we consider the integral equation  b ϕ(x) = λ K(x, s)ϕ(s)[dσ(s) + M dτ (s)], (136) a

where τ (x) is defined by the condition that for every continuous function F (x) (a ≤ x ≤ b)  b F (x)dτ (x) = F (c). a

Equation (136) can be rewritten as  b ϕ(x) − λ K(x, s)ϕ(s) dσ(s) = λM ϕ(c)K(x, c).

(137)

a

If ϕ(x) (≡ 0) is a solution to (136) for some value of λ, then by Theorem 5 we have ϕ(c) = 0. Equation (136), or, what is the same, (137), cannot have eigenvalues in common with (125). In fact, were (125) to have a non-trivial solution ϕ ≡ 0 when λ = λj , then by multiplying each term of this solution by ϕj (s)dσ(s) and integrating over the interval (a, b), we would obtain  b 0 = λj M ϕ(c) K(c, s)ϕj (s)dσ(s) = M ϕ(c)ϕj (c), a

which is impossible. Consequently, when λ is an eigenvalue of the equation (137), we can solve this equation with respect to ϕ, expressing ϕ in terms of the right hand side f (x) = λM ϕ(c)K(x, c). We then obtain ϕ(x) = λM ϕ(c)Γ(x, c; λ).

(138)

Put x = c, and divide through by ϕ(c); we then obtain an equation for the eigenvalues of the integral equation (136): 1 = λM Γ(c, c; λ).

(139)

It is easy to see that, conversely, any root λ = λ(M ) of the equation (139) is an eigenvalue of (136) and to it corresponds, up to a constant factor, the unique fundamental function ϕ(x) = const. Γ(x, c; λ(M )). Equation (139) can be also rewritten as ∞  ϕ2j (c) 1 − = 0. λj − λ M λ j=0

Denoting the right hand side of this equation by Ψ(λ) we find that: ∞ dΨ  ϕ2j (c) 1 + >0 = 2 dλ (λj − λ) M λ2 j=0

(140)

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IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

and Ψ(λ) > 0 for

− ∞ < λ < 0;

Γ(λj − 0) = +∞,

Γ(λi + 0) = −∞

(j = −1, 0, 1, 2, . . . ; λ−1 = 0). We conclude therefore that inside each of the intervals λj−1 < λ < λj (j = 0, 1, 2, . . . ) the function Ψ(λ) has exactly one zero, and has no other zeros. Denoting by λ0 (M ), λ1 (M ), . . . the successive roots of the equation (140) (i.e., the successive eigenvalues of the integral equation (136)), we have 0 < λ0 (M ) < λ0 < λ1 (M ) < λ1 < . . . Since it follows from (140) that dλ 1 =− < 0, dM λM 2 Ψ (λ) we see that the λj (M ) (j = 0, 1, 2, . . . ) are decreasing functions of M 26 . A simple analysis of (140) shows also that lim λj (M ) = λj ,

M →0

lim λj (M ) = λ∗j−1

M →∞

(j = 0, 1, 2, . . . ; λ∗−1 = 0).

Thus, as M increases from 0 to ∞, the number λj (M ) decreases from λj to λ∗j−1 (j = 0, 1, 2, . . . ). (M )

The fundamental function ϕj (x) corresponding to λj (M ) is obtained from (138), with λj (M ) j = 0, 1, 2, . . . ) in place of λ(M ). Since the fundamental function (M ) ϕj (x) has exactly j nodes between a and b and no other zeros, we conclude that the resolvent Γ(x, c; λ) has exactly j nodes between a and b when λ∗j−1 < λ < λj (j = 0, 1, 2, . . . ). This proves the theorem. Remark 1. In this theorem one can replace the function Γ(x, c; λ) by the function D(x, c; λ) = D(λ)Γ(x, c; λ) and then the condition λ = λj (j = 0, 1, 2, . . . ) can be discarded. To be specific, let us assume that c = b, and consequently λ∗j (j = 0, 1, 2, . . . ) are the successive roots of the equation D(b, b; λ) = 0. Let us trace the variation of the minor D(x, b; λ) as a function of x, as λ varies from 0 to ∞. When 0 ≤ λ < λ∗0 , D(x, b; λ) > 0

(a < x ≤ b).

When λ = λ∗0 the minor has a zero at the point x = b; upon further increase of λ, this zero changes into a node α1 (λ), which moves in some manner between a and b: a < α1 (λ) < b (λ∗0 < λ < +∞). 26 As

the mass of the continuum increases, its frequencies decrease.

6. OSCILLATORY PROPERTIES OF FORCED VIBRATIONS

219

This node will be the only interior zero of D(x, b; λ) as long as λ < λ∗1 ; when λ = λ∗1 a new zero appears in the minor D(x, a; λ) at x = b, and this zero transforms to a node α2 (λ) which moves between α1 (λ) and b as λ increases further: α1 (λ) < α2 (λ) < b. These two nodes will be the only interior zeros of D(x, b; λ) as long as λ < λ∗2 , etc.

Thus, when λ passes through λ∗j−1 (j = 1, 2, . . . ), a node αj (λ) is born at the point b, and as long as λ varies within the limits λ∗j−1 < λ < λ∗j , the only interior zeros of the function D(x, b; λ) are the nodes (a 0), then by Theorem 10 D(a, b λ) > 0

(λ > 0).

(141)

This statement can be also obtained directly from the formula     ∞  (−λ)n a s 1 . . . sn ··· K dσ(s1 ), . . . , dσ(sn ), D(a, b; λ) = K(a, b)+ b s1 . . . sn n! n=1 since K(a, b) ≥ 0,

 n

(−1) K

a b

s1 s1

... ...

sn sn



 =K

a s1

s1 .

... ...

. sn

sn b

 ≥0

and, in the case K(a, b) = 0,   a s1 K = K(a, s1 )K(s1 , b) > 0 for a < s1 < b. s1 b Inequality (141) has the following mechanical meaning: If both ends of the continuum S are movable, then in the forced oscillation that arises under the influence of a pulsating force applied at one endpoint, the amplitude of the second endpoint is always different from zero. Remark 2. It is worth to emphasize that while proving Theorems 10 and 10 we have shown as well that a forced oscillation of a continuum S under the influence of a concentrated pulsating force coinsides with a natural harmonic oscillation of the continuum, either when it is loaded with an additional mass at a point x = c, or when it is elastically supported at this point, depending on the interval in which the frequency of the disturbing force lies.

220

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

Remark 3. In proving that the resolvent Γ(x, c; λ) has exactly j nodes between a and b and no other zeros at the movable points when λ∗j−1 < λ < λj we have not used anywhere the fact that c = a or b, but only the fact that ϕj (c) = 0. We recall, as it was explained in Theorem 7 of Section 5, that when ϕj (c) = 0 (and only in this case) we have λ∗j−1 = λj . It is very plausible that for an interior point c (a < c < b) the number of zeros in I of the resolvent Γ(x, c; λ), for λj < λ < λ∗j always fluctuates between j and j + 1, if a zero anti-node is counted as a double zero. However, we do not see how this conjecture could be verified with the methods given above. Thus it is very interesting that in the case of a string, as will be shown in the next section, Theorem 10 remains completely in force, independently of a movable point (interior or endpoint) where a pulsating force is applied. Concluding this section, we note that the propositions of Section 8 of Chap. II are closely related with Theorem 10 ; in particular, they contain a generalization of this theorem to the case of a non-symmetric kernel in the special case that the function σ(x) has a finite number of points of growth. 7. Oscillations of an elastically supported string 1. As we know (Section 7 of Chap. II), the influence function of a string always belongs to the class of single-pair kernels K(x, s), which have the form  ψ(x)χ(s), (x ≤ s) K(x, s) = (142) ψ(s)χ(x), (x ≥ s), where ψ(x) and χ(x) are continuous functions. The following is direct corollary of Theorem 12 of Chap. II: Criterion A. In order for a single-pair kernel K(x, s) (a ≤ x, s ≤ b) to be oscillatory, it is necessary and sufficient that the following two conditions be satisfied: 1◦ ψ(x)χ(x) > 0 when a < x < b. 2◦ The function ψ(x)/χ(x) increases in the interval (a, b). We have already used this criterion to prove that the influence function of a string fastened in the usual manner is oscillatory. We observe that conditions 1◦ and 2◦ are equivalent to the following two conditions: 1 . K(x,  x) > 0  (a < x < b), x 1 x2  > 0 (a < x1 < x2 < b). 2.K x1 x2 In Section 6 it was shown that if a kernel K(x, s) is the influence function of a continuum and if x = c is a movable point of this continuum, then for every A > 0 the kernel    K(x, s) 1 K(x, c)   (143) LA (x, s) = A + K(c, c)  K(c, s) K(c, c) + A  will be the influence function of the continuum S ∗ , which is obtained from S by introducing an elastic support (spring) at the point x = c, with stiffness κ = 1/A. In view of this, the following proposition is of interest (compare with Lemma 7 of Section 6).

7. OSCILLATIONS OF AN ELASTICALLY SUPPORTED STRING

221

Lemma 8. If K(x, s) (a ≤ x, s ≤ b) is a single-pair oscillatory kernel, then so is the kernel LA (x, s) for every A > 0 and every point c (a ≤ c ≤ b). Proof. In view of Lemma 7 of Section 6, this proposition is of interest only for the case of an interior point c (a < c < b). By direct calculations we can verify that if the kernel K(x, s) has the form (142), then  ψ1 (x)χ1 (s) (x ≤ s) LA (x, s) = ψ1 (s)χ1 (x) (x ≥ s), where

& ψ1 (x) =

and

ψ(x) (a ≤ x ≤ c), 1 [(A + ψ(c)χ(c))ψ(x) − ψ 2 (c)χ(x)] (c ≤ x ≤ b), A

⎧ χ2 (c) ⎪ ⎪ ⎨ χ(x) − A + ψ(c)χ(x) ψ(x) χ1 (x) = ⎪ Aχ(x) ⎪ ⎩ A + ψ(c)χ(c) On the other hand, according to (131) of Section   c x 1 . . . xn   K c x1 . . . xn x 1 x2 . . . xn LA +A = x1 x2 . . . xn A + K(c, c)

(a ≤ x ≤ c), (c ≤ x ≤ b). 6, K



x 1 x2 . . . xn x1 x2 . . . xn A + K(c, c)

 ,

from which it is clear that conditions 1 and 2 for the kernel LA (x, s) are also satisfied. It is also easy to verify that the kernel LA (x, s) is oscillatory, by direct verification that conditions 1◦ and 2◦ hold. From this lemma follows Theorem 11. The oscillatory character of the influence function of a string is retained when any number of elastic supports of any stiffness is introduced at movable points of the string. Proof. Indeed, assume that we introduce elastic supports of stiffness κ1 = −1 −1 A−1 1 , κ2 = A2 , . . . , κn = An at the movable points c1 , c2 , . . . , cn of the string S. If K0 (x, s) is the influence function of the original string, and Ki (x, s) the influence function of the string after elastic supports of stiffnesses κ1 = A−1 1 , . . . , κi = (i = 1, 2, . . . , n) are introduced at c , c , . . . , c , then each kernel K A−1 1 2 i i (x, s) will i be related to the preceding one by a relation of type (143), and consequently, according to the lemma, all kernels K1 (x, s), . . . , K(x, s) will be oscillatory single-pair kernels, together with the kernel K0 (x, s). 2. Theorem 11 remains in force under a much broader interpretation of the words “any number” in its formulation. In particular, it remains in force for a string which is secured to a solid elastic base and in addition has “concentrated” elastic supports at some points. We are led to this broader interpretation of the theorem by the following considerations.

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IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

Let K(x, s) be the influence function of a continuum S (S need not necessarily be a string). If we introduce at the points c1 , c2 , . . . , cn elastic supports of stiffnesses κ1 , κ2 , . . . , κn , then the deflection y(x) of the continuum S is given by the formula  n  K(x, s)dQ(s) − Rj K(x, cj ), y(x) = S

j=1

where dQ(s) is the load on the element ds, and R1 , R2 , . . . , Rn are the reactions of the supports at the points c1 , c2 , . . . , cn and, consequently, Rj = κj y(cj ) Thus,

(j = 1, 2, . . . , n).

 K(x, s)dQ(s) −

y(x) = S

n 

κj K(x, cj )y(cj ).

(144)

j=1

Assigning here values c1 , c2 , . . . , cn to x, we obtain a system of equations from which we determine the deflection y(cj ) (j = 1, 2, . . . , n), and then we obtain also y(x). If we introduce the piecewise constant function τ (x), which has jumps at the points c1 , c2 , . . . , cn , with jumps respectively equal to κ1 , κ2 , . . . , κn , then (144) can be written as an integral equation for y(x):   K(x, s) dQ(s) − K(x, s)y(s) dτ (s). (145) y(x) = S

S

In the general interpretation of the notion of elastic support of a string, we can imagine τ (x) to be an arbitrary non-decreasing function, so that as the string is deflected, some device exercises an elastic reaction force in the opposite direction to the deflection y(x), equal to y(s)dτ (s) for each element ds, so that the deflection under the action of the active forces dQ(s) is determined from equation (145). Equation (145) is a special case of a loaded integral equations  y(x) = λ K(x, s)y(s) dτ (s) + f (x), (146) S

corresponding to

 λ = −1,

K(x, s)dQ(s).

f (x) =

(147)

S

When λ is not an eigenvalue, equation (146) is solved by  y(x) = f (x) + λ Γ(τ ) (x, s; λ)f (s)dτ (s),

(148)

S

where Γ(τ ) (x, s; λ) is the resolvent of (146). Introducing into (148) the values of λ and f (x) from (147) and taking λ = −1 in the functional equation of the resolvent,  Γ(τ ) (x, s; λ) = K(x, s) + λ Γ(τ ) (x, r; λ)K(r, s)dτ (r), S

8. FORCED OSCILLATIONS OF A STRING

223



we find that

Γ(τ ) (x, s; −1)dQ(s),

y(x) =

(149)

S

and we conclude that the influence function of an elastically supported continuum is the resolvent Γ(τ ) (x, s; −1) of (146). We shall show in the next section that the resolvent Γ(τ ) (x, s; λ) of a single-pair kernel K(x, s) is always a single-pair kernel, for every distribution function τ (s). If in addition K(x, s) is an oscillatory kernel, then Γ(τ ) (x, s; λ) will be an oscillatory (τ ) kernel for every λ less than the smallest eigenvalue λ0 of (146), and, in particular, for all negative λ. The validity of Theorem 11 will follow from this for the most general case of elastically supported string. 8. Forced oscillations of a string 1. Given a single-pair kernel K(x, s), we form a new kernel    K(x, s) K(x, c)  1 ,  K ∗ (x, s) = K(c, c)  K(c, s) K(c, c)  where c is an interior point: a < c < b. Then the kernel K ∗ (x, s) will have the following property K ∗ (x, s) = K ∗ (s, x) = 0 for x < c < s.

(150)

∗

If K(x, s) is the influence function of a string S, then K (x, s) will be the influence function of the string S ∗ obtained from S by introducing a stationary support at the point c. But when this support is introduced, the string S is divided into two parts: S1 and S2 , which are deflected independently of each other, as expressed by (150). Let p0 < p1 < p2 < . . . be the successive frequencies of the natural oscillations of the string S, and p∗0 ≤ p∗1 ≤ p∗2 ≤ . . . the successive frequencies of the natural oscillations of S ∗ , i.e., the frequencies of the segments S1 , S2 , arranged in a single sequence in increasing order. According to the remark to Theorem 7 of Section 5, we have the inequalities p0 ≤ p∗0 ≤ p1 ≤ p∗1 ≤ p2 ≤ p∗2 ≤ . . . .

(151)

If c does not coincide with any of the nodes of the natural oscillations of S, then (see Section 5, Theorem 7, and the remark which follow): p0 < p∗0 < p1 < p∗1 < p2 < . . . . Thus, in this case, the segments S1 and S2 cannot have common frequencies. This is understandable: were S1 and S2 to have a common frequency p∗ , then by “normalizing” in a suitable manner the harmonic oscillation of the segments S1 and S2 with the frequency p∗ , we could arrange a harmonic oscillation of all of S ∗ without a “break” at the point c, which simultaneously would be a natural oscillation of S with a node at the point c. To the contrary, if we place the support at a node of a natural oscillation of S with frequency p, then it remains an oscillation of S ∗ and thus generates natural oscillations of S1 and S2 . Taking (151) into account, we then have p∗j−1 = pj = p∗j .

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IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

Thus p0 < p∗0 always holds, and p∗j−1 ≤ pj ≤ p∗j

(j = 1, 2, . . . );

furthermore in each such relation either all three quantities are different, or all three are equal, the latter taking place if and only if the support is placed at a node of the oscillation with frequency p∗j .27 2. Theorem 10 admits the following generalization for the case of a string: Theorem 12. If a movable point c is not a node of a natural oscillation of a given frequency pj (j = 0, 1, 2, . . . ) of the string S, then the forced oscillation, under the influence of a pulsating force of frequency p at the point c, where p∗j−1 < p < p∗j ,

p = pj ,

has exactly j nodes. Proof. As we know (see Remark 2 of Section 6) the amplitude function Γ(x, c; p2 ) is simultaneously the j-th amplitude function of the string S, either loaded with an additional mass at the point c (if p∗j−1 < p < pj ), or else elastically supported at the point c (if pj < p < p∗j ). In either case (by Theorem 11) this amplitude function will be the j-th fundamental function of a loaded integral equation with an oscillatory kernel. From this we obtain the theorem. Analyzing the mathematical contents of the proof of Theorem 12, we readily verify that we have essentially proved a general theorem concerning the resolvent of an integral equation with a single-pair oscillatory kernel. The formulation of this theorem involves no difficulty; we do not give it here, since the property it expresses is a special case of the deeper properties of the resolvent, given in the next section (see Theorems 14 and 15, and formula (164)). In particular, it will follow from Theorem 14 of the next section that in a forced oscillation of a string under the action of a pulsating force applied at the point c, the nodes located to the left and to the right of c move monotonically from the point c towards the end of the interval as the frequency of pulsation p increases; as the frequency p passes through the values p∗j , a new node is born at the point c28 which moves to the left or to the right end of the string, depending on whether p∗j is a 27 From this, we can prove the following proposition for a string in a manner similar to the proof of Theorem 8 for a rod: Let a string S be divided by introducing n intermediate supports into n + 1 parts Sk (k = (k) 0, 1, . . . , n) with fundamental tones p0 (k = 0, 1, . . . , n). Then the n-th overtone pn of the string S will satisfy the inequalities: (0)

(1)

(n)

(0)

(1)

(n)

min(p0 , p0 , . . . , p0 ) ≤ pn ≤ max(p0 , p0 , . . . , p0 ), where equality in the left or in the right occurs if and only if the supports are placed at the nodes of the n-th overtone S, so that (0)

p0

(1)

= p0

(n)

= · · · = p0

= pn .

This proposition can be suitably generalized to all other overtones pn+j (j = 1, 2, . . . ) (see Theorem 8). 28 If this point is not a node of the j-th overtone (p∗ = p ); in the opposite case two nodes j j will be born at the point c, moving in opposite directions.

9. THE RESOLVENT OF AN OSCILLATORY SINGLE-PAIR KERNEL

225

frequency of the left or of the right part of the string (when p = p∗j the right or the left part of the string remains at rest in the forced oscillation). 9. The resolvent of an oscillatory single-pair kernel 1. Let us show first that in a loaded integral equation with continuous singlepair kernel  ψ(x)χ(s) (x ≤ s) K(x, s) = ψ(s)χ(x) (x ≥ s), the resolvent Γ(x, s; λ) is also a single-pair kernel. Using the kernel V (x, s) = ψ(x)χ(s) − ψ(s)χ(x)

(a ≤ s ≤ x ≤ b),

the single-pair kernel K(x, s) can be represented in the form  ψ(x)χ(s) (x ≤ s), K(x, s) = ψ(x)χ(s) − V (x, s) (x ≥ s). By this representation, the integral equation  b K(x, s)ϕ(s)dσ(s) + f (x) ϕ(x) = λ

(152)

a

is equivalent to the system of two equations:  x ⎫ ⎪ V (x, s)ϕ(s)dσ(s) = f (x) + Ωψ(x)⎪ ϕ(x) + λ ⎬ a



⎪ ⎪ ⎭

b

χ(s)ϕ(s)dσ(s) = Ω.

λ

(153)

a

The Volterra equation:



x

ϕ(x) + λ

V (x, s)ϕ(s)dσ(s) = F (x),

(154)

a

for any continuous function F (x), has a unique solution, which can be expressed in the form29  x ϕ(x) = F (x) − λ V (x, s; λ)F (s)dσ(s), (155) a

where V (x, s; λ) =

∞ 

(−λ)n V (n+1) (x, s),

(156)

n=0

with V (1) (x, s) = V (x, s),  x (n) V (x, r)V (n−1) (r, s)dσ(r) V (x, s) =

(n = 2, 3, . . . ).

s

Since V (s, s) ≡ 0 (a ≤ s ≤ b), then, as can be readily seen, all kernels V (n) (x, s) (n = 1, 2, . . . ) are continuous and satisfy the condition V (n) (s, s) ≡ 0 (a ≤ s ≤ b). 29 See

Privalov [43], p. 31.

226

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

On the other hand, since for every λ, the series (156) converges uniformly in a triangle a ≤ s ≤ x ≤ b30 then the resolvent V (x, s; λ) (a ≤ s ≤ x ≤ b) is continuous and V (s, s; λ) ≡ 0 (a ≤ s ≤ b). From this we conclude that the solution ϕ(x) of the equation (154) is always continuous when F (x) is continuous. Using (155) to solve the first equation of (153), we obtain  x ϕ(x) = f (x) − λ V (x, s; λ)f (s)dσ(s) + Ωψ(x; λ), (157) a



where

x

ψ(x; λ) = ψ(x) − λ

V (x, s; λ)ψ(s)dσ(s).

(158)

a

Substituting then the expression (157) for ϕ(x) in the second equation of (153) and solving it with respect to Ω, we get: λ Ω=

$b a

1−λ

f (s)χ(s; λ)dσ(s)

$b a

where

,

(159)

ψ(x; λ)χ(x)dσ(x) 

b

χ(s; λ) = χ(s) − λ

V (x, s; λ)χ(x)dσ(x).

(160)

s

We note that the continuous functions ψ(x; λ) and χ(s; λ) are solutions to the Volterra equations:  x ⎫ ⎪ ψ(x; λ) = ψ(x) − λ V (x, s)ψ(s; λ)dσ(s),⎪ ⎬ a (161)  b ⎪ ⎪ V (x, s)χ(x; λ)dσ(x). ⎭ χ(s; λ) = χ(s) − λ s

In addition, by (158) and (160),  b  ψ(s; λ)χ(s)dσ(s) = a

b

ψ(s)χ(s; λ)dσ(s).

(162)

a

Let ξ be an arbitrary fixed number in the interval a < x < b. For f (x) = K(x, ξ) the solution to (152) will be the resolvent Γ(x, ξ; λ). Calculating the resolvent for the values x ≤ ξ by (157) and taking into account that now f (s) = ψ(s)χ(ξ) for s ≤ ξ, we obtain Γ(x, ξ; λ) = [Ωξ + χ(ξ)]ψ(x; λ) 30 The

(x ≤ ξ),

series (156) is majorized by the following numerical series ∞  n=0

1 M n [σ(b) − σ(a)]n , n!

where M =

max

a≤s≤x≤b

|V (x, s)|.

(163)

9. THE RESOLVENT OF AN OSCILLATORY SINGLE-PAIR KERNEL

227

where, according to (159) and (162), λ

$b

K(s, ξ)χ(s; λ)dσ(s)

a

Ωξ =

1−λ

$b

ψ(x; λ)χ(x)dσ(x)

a

$ξ $b χ(ξ) ψ(s)χ(s; λ)dσ(s) + ψ(ξ) χ(s)χ(s; λ)dσ(s) =λ

a

ξ

1−λ

$b

,

ψ(s)χ(s; λ)dσ(s)

a

and, consequently, χ(ξ) − λ

$b

V (s; ξ)χ(s; λ)dσ(s)

ξ

Ωξ + χ(ξ) = 1−λ

$b

ψ(s)χ(s; λ)dσ(s)

a

χ(ξ; λ)

= 1−λ

$b

.

ψ(s)χ(s; λ)dσ(s)

a

Thus, in the loaded integral equation (152) with a single-pair kernel, the resolvent Γ(x, s; λ) = Γ(s, x; λ) is also a single-pair kernel, namely Γ(x, s; λ) =

ψ(x; λ)χ(s; λ) $b 1 − λ ψ(x; λ)χ(x)dσ(x)

(x ≤ s),

(164)

a

where the functions ψ(x; λ) and χ(s; λ) are determined from the adjoint Volterra equations (161) with kernel V (x, s) = ψ(x)χ(s) − ψ(s)χ(x). It turns out that the denominator in (164) is nothing but the Fredholm determinant D(λ) of the equation (152): 

b

D(λ) = 1 − λ

ψ(s; λ)χ(s)dσ(s),

(165)

a

and consequently, the numerator in (164) is the minor D(x, s; λ):  ψ(x; λ)χ(s; λ) (x ≤ s), D(x, s; λ) = ψ(s; λ)χ(x; λ) (x ≥ s).

(166)

To obtain (165) we notice that, according to the rule of calculating the determinant of a single-pair matrix (see Section 3, Chap. II)   s 1 s2 . . . sn = ψ(s1 )V (s2 , s1 )V (s3 , s2 ) . . . V (sn , sn−1 )χ(sn ) K s1 s2 . . . sn (a ≤ s1 ≤ s2 ≤ · · · ≤ sn ≤ b;

n = 2, 3, . . . ),

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IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

and consequently, by (5 )  b  c1 = K(s, s)dσ(s) = 

a



a

b



sn

cn =

 ···





ψ(s)χ(s)dσ(s), a

s2



K

a

b

s3

b

a

a

s 1 s2 . . . sn s1 s2 . . . sn

 dσ(s1 )dσ(s2 ) . . . dσ(sn )

x

χ(x)V (n−1) (x, s)ψ(s)dσ(s)dσ(x)

= a

(n = 2, 3, . . . );

a

hence, according to (5) and (156), D(λ) = 1 +

∞ 

cn (−λ)n =

n=1



=1−λ

b

 χ(x) ψ(x) − λ

a



x

V (x, s; λ)ψ(s)dσ(s) dσ(x),

a

which implies (165). Incidentally, we could have obtained (166) by the same method, calculating directly the coefficients of the expansion of D(x, s; λ) in powers of λ. 2. From the expansion of the resolvent ∞

D(x, s; λ)  ϕj (x)ϕj (s) = , D(λ) λj − λ j=0 we obtain31 taking (166) into account: ϕj (x)ϕj (s) =

ψ(x; λj )χ(s; λj ) D (λj )

(a ≤ x ≤ s ≤ b; j = 0, 1, 2, . . . ).

(167)

From this we conclude that there exist such constants cj and dj (j = 0, 1, 2, . . . ) that ϕj (x) = cj ψ(x; λj ) = dj χ(x; λj ) (j = 0, 1, 2, . . . ). (168) 3. We can now prove without difficulty the following Theorem 13. In a loaded integral equation (152) with a single-pair oscillatory kernel, the resolvent Γ(x, s; λ) is also a single-pair oscillatory kernel for all values of λ less than the smallest eigenvalue λ0 of (152). 31 (167)

is obtained under the assumption that λj is a simple eigenvalue. For a single-pair kernel this is always true, for if a certain eigenvalue λj were to have a multiplicity greater than one, we would have instead of (167) D(x, s; λj ) ≡ ψ(x; λj )χ(s; λj ) ≡ 0

(a ≤ x ≤ s ≤ b),

from which we would readily conclude the existence of such a c (a ≤ c ≤ b), that ψ(x; λj ) = 0

(a < x < c),

χ(x; λj ) = 0

(c < x < b).

From this and (161) we obtain ψ(x) = 0 when x ≤ c and χ(x) = 0 when x ≥ c and then K(x, s) ≡ 0 (a ≤ x, s ≤ b).

9. THE RESOLVENT OF AN OSCILLATORY SINGLE-PAIR KERNEL

229

Proof. Since the single-pair property of the kernel Γ(x, s; λ) has already been established, to prove that it is oscillatory for λ < λ0 according to Criterion A of Sect. 7, it remains to show that when λ < λ0 the quadratic form Φ=

m 

(a < x1 < x2 < · · · < xm < b)

Γ(xi , xk ; λ)ξi ξk

j,k=1

is positive32 . When 0 < λ < λ0 we have the expansion  m ∞   m   (n+1) K(xi , xk )ξi ξk + K (xi , xk )ξi ξk λn , Φ= n=1

i,k=1

i,k=1

in which the first term is positive (because the kernel K(x, s) is oscillatory) and all remaining terms are known to be non-negative. When −∞ < λ < 0 we have D(λ) > 0, and D(λ) · Φ =

m 

D(xi , xk ; λ)ξi ξk =

i,k=1

+

m 

K(xi , xk )ξi ξk

i,k=1

    b  b  ∞ m  (−λ)n xi s 1 . . . s n ξi ξk dσ(s1 ) . . . dσ(sn ), ··· K xk s 1 . . . s n n! a a n=1 i,k=1

where the first term is positive and all the rest are non-negative, since for every s1 , s2 , . . . , sn in (a, b),   m  xi s 1 . . . s n ξi ξk ≥ 0 K (n = 1, 2, . . . ). xk s 1 . . . s n i,k=1

This proves the theorem. As it was explained in Section 7, this theorem leads to a broader interpretation of Theorem 11. 4. Now we investigate the oscillatory properties of the function ψ(x; λ) in the interval a ≤ x ≤ b for various values of λ. For this purpose we notice the following property of the loaded integral equation (152) with a single-pair kernel: 1◦ If χ(b) = 0, then ψ(b; λ) = ψ(b)D(λ). (169) Indeed, if χ(b) = 0, we have V (b, s) = ψ(b)χ(s), and consequently,  ψ(b; λ) = ψ(b) − λ 

b

V (b, s)ψ(s; λ)dσ(s) = a



= ψ(b) 1 − λ

b

ψ(s; λ)χ(s)dσ(s) . a

32 According

to criterion A, it is sufficient to show this for the values m = 1 and 2.

230

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

Let now a < ξ < b, and let us consider the integral equation  ξ Kξ (x, s)ϕ(s)dσ(s), ϕ(x) = λ

(170)

a

   K(x, s) K(x, ξ)  1 .  Kξ (x, s) = K(ξ, ξ)  K(ξ, s) K(ξ, ξ)  It is easy to see that  ψ(x)χξ (s) (a ≤ x ≤ s ≤ ξ) Kξ (x, s) = ψ(s)χξ (x) (a ≤ s ≤ x ≤ ξ),

where

where χξ (x) = χ(x) −

χ(ξ) ψ(x) ψ(ξ)

(a ≤ x ≤ ξ).

Noticing that Vξ (x, s) = ψ(x)χξ (s) − ψ(s)χξ (x) = ψ(x)χ(s) − ψ(s)χ(x) = V (x; s), we find that the definition of the function ψ(x, λ) in the interval a ≤ x ≤ ξ remains the same for (170) as for (152). Since χξ (ξ) = 0, we can use formula (169) as applied to the function ψ(ξ; λ) and to the Fredholm determinant Dξ (λ) of (170). Thus ψ(ξ; λ) = ψ(ξ)Dξ (λ). If K(x, s) (a ≤ x, s ≤ b) is an oscillatory kernel, the kernel Kξ (x, s) (a ≤ x, s ≤ ξ) will also be oscillatory. In this case Dξ (λ) will have only simple positive zeros: (0 0. In this case, the preceding arguments apply with ξ = b. According to the remark in Section 5 λn−1 (b) < λn < λn (b) (n = 0, 1, 2, . . . ).

(175)

Let us use (171). By this relation and (168),  ∞   λn . 1− ϕn (ξ) = const. λj (ξ) j=0 As ξ decreases from b to a, the function ϕn (ξ) changes sign exactly n times and whenever it does some λj (ξ) must pass through the value λn . By (175) we conclude that λ0 (ξ), λ1 (ξ), . . . , λn−1 (ξ) will pass through the value λn , and no other λj (ξ) will. Consequently, when ξ is sufficiently close to a, we have λ0 (ξ) > λn and since λn → +∞ as n → ∞, then (174) is proved for the case under consideration. Whenever the initial interval (a, b) is replaced by some other interval (a, b ), where a < b < b, the function ψ(x; λ) remains unchanged in the interval a ≤ x ≤ b , and χ(b ) = 0, so that (174) is always satisfied. This proves Proposition 2◦ . Very little need to be added to the foregoing arguments to obtain the following theorem.

232

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

Theorem 14. If K(x, s) is a single-pair oscillatory kernel and ψ(b) = 0, then when λn−1 (b) < λ ≤ λn (b) (n = 0, 1, 2, . . . ; λ−1 = −∞) (176) the function ψ(x; λ) has exactly n zeros inside (a, b): α1 (λ) < α2 (λ) < · · · < αn (λ). These zeros are nodes, they decrease with increasing λ, and the numbers λn (b) (n = 0, 1, 2, . . . ) are the successive roots of the equation ψ(b; λ) = 0.

(177)

Proof. Without loss of generality one can assume χ(b) = 0, for otherwise we would replace χ(x) by χ1 (x) = ψ(x) + χ(x) for which we already have χ1 (b) = 0 in the definition of the kernel K(x, s); then the kernel will remain oscillatory, and the function ψ(x; λ) will not change. Consequently, the representation of (171), as was already noted, will hold for all values ξ ∈ (a, b]. If λ satisfies inequality (176), then of all λj (ξ) (j = 0, 1, 2, . . . ) only λ0 (ξ), λ1 (ξ), . . . , λn−1 (ξ) while increasing will pass through the value λ as ξ decreases from b to a. It follows from this that the function ψ(x; λ), for the indicated values of λ, has exactly n nodes α1 (λ), α2 (λ), . . . , αn (λ) inside (a, b), and no other zeros inside (a, b). Furthermore, all these nodes are the roots of the corresponding equations λj (ξ) = λ (j = 0, 1, 2, . . . , n − 1). From Proposition 2◦ also follows that the functions αj (λ) (j = 0, 1, 2, . . . ) decrease with increasing λ. Remark 1. Since D(x, b; λ) = χ(b) ψ(x; λ), Theorem 14 implies the corresponding property of the resolvent Γ(x, b; λ) in the case that K(b, b) = 0. Applied to the resolvent of a string, this yields the statement made at the end of Section 8. Remark 2. If ψ(b) = 0, then χ(b) = 0 (by the monotonicity of the ratio ψ(x)/χ(x)) and consequently V (b, s) = 0 (a ≤ x ≤ b) and



b

V (b, s)ψ(s; λ)dσ(s) ≡ 0 (−∞ < λ < +∞).

ψ(b; λ) = ψ(b) + λ a

Thus, the numbers λj (b) (j = 0, 1, 2, . . . ) cannot be defined as roots of the equation (177), but they can be obtained as limits: λj (b) = lim λj (ξ) ξ→b

(j = 0, 1, 2, . . . ).

(178)

Since the positive ratio χ(x)/ψ(x) is a decreasing function of x, there exists a limit χ(x) , B = lim x→b ψ(x) and consequently, regardless of whether K(b, b) = 0 or = 0, the oscillatory kernel Kb (x, s) is always well-defined as Kb (x, s) = lim Kξ (x, s) = K(x, s) − Bψ(x)ψ(s) ξ→b

(a ≤ x, s < b).

9. THE RESOLVENT OF AN OSCILLATORY SINGLE-PAIR KERNEL

233

It can be readily seen that the numbers λj (b) (j = 0, 1, 2, . . . ) defined by (178) will be the eigenvalues of the integral equation  b ϕ(x) = λ Kb (x, s)ϕ(s)dσ(s). (179) a

From all the foregoing it is easy to conclude that with the modified definitions of the numbers λj (b) (j = 0, 1, 2, . . . ) the theorem also remains in force when ψ(b) = 0. 5. It is obvious that χ(x; λ) has properties analogous to those of ψ(x; λ). Thus to χ(x; λ) corresponds a sequence μn (n = 0, 1, 2, . . . ) such that when μn−1 < λ ≤ μn

(n = 0, 1, 2, . . . ; μ−1 = −∞)

this function has inside (a, b) exactly n zeros, all of them are nodes, β1 (λ) > β2 (λ) · · · > βn (λ), and they increase with increasing λ. If χ(a) =  0, the numbers μn are defined as successive roots of the equation χ(a; λ) = 0. Regardless of whether χ(a) = 0 or = 0, these numbers can be defined as the eigenvalues of the integral equation which is obtained from (179) by replacing the kernel Kb (x, s) by the kernel Ka (x, s) = K(x, s) − Aχ(x)χ(s) where A = lim

x→a

(a ≤ x, s ≤ b),

ψ(x) . χ(x)

We have Theorem 15. When λ is one of the eigenvalues of the integral equation (152), the nodes of the functions ψ(x; λ) and χ(x; λ) coincide; for other values of λ the nodes of these functions alternate. Proof. The first statement follows directly from formulas (168). Taking into account the monotonicity of the nodes α(λ) and β(λ), to prove the second statement it is enough to show that when λ is not λj (j = 0, 1, 2, . . . ) the functions ψ(x; λ) and χ(x; λ) cannot have a common zero in (a, b). Let us now assume the contrary, i.e., that for some λ = λ0 (= λj ; j = 0, 1, 2, . . . ) and a some α(a < α < b) we have ψ(α; λ0 ) = χ(α; λ0 ) = 0. Then the function F (λ) =

∞ ψ(α; λ)χ(α; λ)  ϕ2j (α) = D(λ) λ −λ j=0 j

has at the point λ0 a zero of at least second order, and this is impossible, since F  (λ0 ) =

∞  j=0

ϕ2j (α) > 0. (λj − λ)2

This proves the theorem. Theorems 14 and 15 can be used to give another proof of Theorem 12.

234

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

From Theorems 14 and 15 also easily follows the statement on the forced oscillations of a string which was made at the end of Section 8. To conclude this section, we notice that the properties of the functions ψ(x; λ) and χ(x; λ) proved here could be also obtained without using the general theory of integral equations with oscillatory kernels, but instead using systematically the fact that conditions 1◦ and 2◦ of criterion A (Section 7) imply that a single-pair kernel is positive definite, and the special structure of its resolvent and the supporting kernels Kξ (x, s). After this one could obtain, independently of the general theory, all the oscillatory properties of the fundamental functions of integral equations with single-pair oscillatory kernels. We have done something similar in the algebraic case, since in Section 1 of Chap. II the theory of the oscillatory properties of Jacobi matrices (which are the inverses of single-pair matrices) was developed independent of the general theory of oscillatory matrices. 10. The Sturm–Liouville equations 1. If a string is acted upon by transverse forces with intensity f (x) (0 ≤ x ≤ l) then the connection between the deflection y and f is expressed by −T y  = f, where T is the tension of the string. If the string is fastened to a certain elastic base which acts on an element dx with the force −k(x)ydx, then the connection between y and f is obtained by replacing f (x) in (180) by the function f (x) − k(x)y(x), which then yields −T y  + k(x)y = f (x). If it is assumed that the distribution of the masses along the string has a density ρ(x), then according to the d’Alembert principle the equation of free oscillations will be ∂2y ∂2y −T + k(x)y = −ρ(x) 2 . (180) 2 ∂x ∂t With respect to the endpoints of the string, we make an assumption that they are fastened with massless rollers which slide freely over ideally smooth contours C0 and C1 . We denote by AB the straight-line position of equilibrium of the string (when only the tension force acts). Thus, the interval AB is perpendicular at the endpoints A and B to the contours C0 and C1 . For small oscillations of a string, starting from the position AB, its deflection y(x, t) will satisfy two boundary conditions, which follow from the mechanical assumption that the forces acting on the rollers with zero masses are balanced (the reaction of the string is balanced by the reaction of the smooth contour). The latter is possible only if the line of deflection of the string remains at all times perpendicular at its ends to the contours C0 and C1 . It is easy to show that in differential form these conditions are written as     ∂y ∂y  − κ0 y  + κ1 y  = 0, = 0, (181) ∂x ∂x x=0 x=l

10. THE STURM–LIOUVILLE EQUATIONS

235

where κ0 and κ1 are the curvatures of the contours C0 and C1 at the points A and B respectively. We shall treat the case of a stationary end as a case in which the corresponding coefficient κ is infinite (the corresponding contour C degenerates to a circle of zero radius). It follows from (180) and (181) that for harmonic oscillations of a string S: y(x, t) = ϕ(x) sin(pt + α), the amplitude function ϕ(x) and the frequency p are determined from the boundaryvalue problem: ' T ϕ − k(x)ϕ + λρ(x)ϕ = 0, (182) ϕ (0) − κ0 ϕ(0) = 0, ϕ (l) − κ1 ϕ(l) = 0 (λ = p2 ). 2. This boundary value problem differs little from the most general regular boundary-value problem of Sturm–Liouville:   ⎫ d dϕ ⎪ p(x) − q(x)ϕ + λr(x)ϕ = 0⎪ ⎪ ⎪ dx dx ⎪ ⎪   ⎪ ⎬ dϕ  sin α · ϕ + cos α · p = 0 (183) dx x=a ⎪ ⎪ ⎪   ⎪ ⎪ dϕ  ⎪ ⎪ ⎭ sin β · ϕ + cos β · p = 0  dx x=b where α and β are real numbers. Regularity here means the non-negativity almost everywhere of r(x), the nonvanishing of p(x) identically in any sub-interval, and the integrability of the three functions r(x), q(x), and p−1 (s) in (a, b). It is obvious that the boundary value problem (182), with integrable functions k(x) and ρ(x), is a special case of (183). On the other hand, the condition of integrability of p−1 (x) permits the change of variable in (183)  x dx s= , (184) p(x) a by which the system (183) is reduced to the form (182) with  b dx , κ0 = − cot α, κ1 = cot β T = 1, l = p(x) a and k(s) = p(x)q(x), ρ(s) = p(x)r(x), and integrability of q(x) and r(x) implies integrability of k(s) and ρ(s). We notice that integrability of the function r(s) is more natural for the string problem, since it means that the total mass of the string is finite. From the purely mathematical point of view the integrability of the functions k(s) and ρ(s) is important because it ensures the existence and uniqueness of a solution to the system d2 ϕ − k(s)ϕ + λρ(s)ϕ = g(s), ds2 ϕ(s0 ) = c0 , ϕ (s0 ) = c1 ,

236

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

for all λ, c0 , c1 , s0 (0 ≤ s0 ≤ l) and g(s). We shall not stop, however, to prove this. In view of all of the foregoing, the only serious difference between systems (182) and (183) lies in the fact that in the system (182), if it corresponds to a mechanical problem, we always have k(x) ≥ 0, χ0 ≥ 0, and χ1 ≥ 0, whereas in the general system (183) no assumptions are made on the signs of the function q(x) and the numbers cot α and cot β. The latter circumstances can cause the problem (183) to have negative eigenvalues. However, as will be shown in Subsection 4 below, any boundary value problem (183) can have only a finite number of negative eigenvalues. Therefore, if we choose a sufficiently large γ and make the substitution λ → λ−γ in (183), then the boundary value problem (183) will be converted into a problem that has only positive eigenvalues. And we can show that a boundary value problem with positive eigenvalues is always equivalent to a loaded integral equation with a single-pair oscillatory kernel. In this way it will be shown that Sturm’s oscillation theorems for the fundamental functions of the boundary value problem (183) are obtained from our general results in a natural way, namely, they follow from the corresponding theorems for integral equations with simplest oscillatory kernels (single-pair ones). 3. Let us denote by L[ϕ] the operator33∗   dϕ d p(x) + q(x)ϕ, L[ϕ] = − dx dx in the system (183). To investigate the boundary value problem (183) we introduce two solutions ψ(x) and χ(x) of the homogeneous equation L[ϕ] = 0, the first of them satisfying the first boundary condition   dψ  sin α · ψ + cos α · p = 0, dx x=a and the second satisfying the second boundary condition   dχ  sin β · χ + cos β · p = 0. dx x=b From

  d dψ dχ p χ− ψ = ψL(χ) − χL(ψ) = 0 dx dx dx

33∗ The operator L[ϕ] will be considered applicable to a function ϕ(x), if ϕ(x) is absolutely continuous, and its derivative ϕ (x), which exists almost everywhere, after being multiplied by p(x), coincides almost everywhere with an absolutely continuous function. In other words, these conditions can be stated as a requirement that the function ϕ(x) (a ≤ x ≤ b), in which the variable x is expressed in terms of s (from (184)), have absolutely continuous derivative with respect to s. For such functions ϕ(x) (a ≤ x ≤ b), we shall understand p dϕ/dx as a continuous function defined by dϕ dϕ = . p dx ds Note that only after this explanation do the boundary conditions in (183), as well as the differential equation itself, acquire a precise meaning.

10. THE STURM–LIOUVILLE EQUATIONS

we obtain the well-known identity   dψ dχ p χ− ψ ≡C dx dx

237

(a ≤ x ≤ b).

The constant C vanishes if and only if the functions ψ and χ are linearly dependent. In this case λ = 0 is an eigenvalue of the boundary value problem (183) (the function ψ satisfies the system (183) when λ = 0) and the boundary conditions in (183) are called singular. Assuming that this is not the case, we multiply ψ (or χ) by C −1 to achieve   dχ dψ χ− ψ ≡1 (a ≤ x ≤ b). (185) p dx dx We then form the kernel



K(x, s) =

ψ(x)χ(s) (x ≤ s) . ψ(s)χ(x) (x ≥ s)

This kernel has an important property: for every integrable function f (x) (a ≤ x ≤ b) it produces a solution ϕ(x) of the system: ⎫ L(ϕ) = f, ⎪ ⎪   ⎪ ⎪ ⎪ dϕ  ⎬ sin α · ϕ + cos α · p = 0  dx x=a (186) ⎪   ⎪ ⎪ dϕ  ⎪ ⎭ sin β · ϕ + cos β · p = 0⎪ dx x=b by the following simple formula 

b

ϕ(x) =

K(x, s)f (s)ds.

(187)

a

Without giving a general definition of the Green function, we notice that because of this property the kernel K(x, s) is called the Green function of the operator L(ϕ), corresponding to the boundary conditions under consideration (or else, the Green function of the boundary value problem (183)). In the case that the system (183) expresses the dependence of the deflection ϕ of a string on the intensity f of the transverse forces, the Green function will be nothing else but the influence function of the string. Formula (187) can also be written as  x  b ψ(x) = χ(x) ψ(s)f (s)ds + ψ(x) χ(s)f (s)ds, a

x

and it is now easy to verify by direct substitution into (186) that it always yields a solution to this system. The uniqueness of the solution for every integrable right hand side f (x) follows from the existence of the unique solution ϕ ≡ 0 of the homogeneous system (when f ≡ 0), according to the non-singularity of the boundary conditions. Since the boundary value problem (183) is obtained from (186) by substituting λr(x)ϕ(x) for f (x), it follows by making a corresponding substitution in (187)

238

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

that this boundary value problem is equivalent to the everywhere loaded integral equation  b ϕ(x) = λ K(x, s)ϕ(s)dσ(s), (188) a

where dσ(x) = r(x)dx. Thus it is important to find out when the Green function is an oscillatory kernel. An answer to this is given by the following theorem Theorem 16. The Green function K(x, s) (a ≤ x, s ≤ b) of a boundary value problem (185) is an oscillatory kernel if and only if K(x, x) = ψ(x)χ(x) > 0

(a < x < b).

(189)

This condition is equivalent to positive definiteness of the Green function or, which is the same, to positivity of all eigenvalues of the boundary value problem (183)34 . Proof. Indeed, since the Green function K(x, s) is a single-pair kernel, then according to criterion A of Section 7, this kernel will be oscillatory if and only if, in addition to (189), the ratio ψ(x)/χ(x) is increasing. The latter condition for the Green function is always satisfied on every interval on which χ(x) = 0, since, according to (185),   d ψ(x) 1 = 2 > 0. dx χ(x) χ (x) This proves the first statement of the theorem. The second statement is obtained from the following two remarks. An oscillatory symmetric kernel is always positive definite. For a single-pair Green function K(x, s) the converse is also true: if it is positive definite then it is oscillatory. Indeed, if K(x, s) is a positive definite kernel, then in any case K(x, x) ≥ 0 for

a0 χ(sk ) χ(sk−1 ) (k = 2, 3, . . . , n).

(192)

Indeed, if there is no zero of χ(x) between sk−1 and sk , then (192) follows from condition 2◦ ; if there is a zero αj between sk−1 and sk , then the numbers sk−1 and

10. THE STURM–LIOUVILLE EQUATIONS

241

sk lie in the interval [αj − ε, αj + ε] and (192) will follow from (191) and from the fact that in this case χ(sk )χ(sk−1 ) < 0. We have proved thereby that the number ν + of negative squares in the form F equals ν or ν + 1, depending on whether Δ1 = K(s1 , s1 ) is positive or negative. Let us remark that if s1 is sufficiently close to a, then the number ν + coincides with the number we wish to prove is equal to N . Let now λ(1) < λ(2) < · · · < λ(m) (< 0)

(193)

be some of the negative eigenvalues of (190) and ϕ(1) (x), ϕ(2) (x), . . . , ϕ(m) (x) the corresponding fundamental functions, so that  b  b ϕ(p) (x) = λ(p) K(x, s)ϕ(p) (s)dσ(s); ϕ(p) (s)ϕ(q) (s)dσ(s) = δpq a

a

(p, q = 1, 2, . . . , m). Then for every function Φ(x) =

m 

ηp ϕ

(p)

 m

(x)

p=1

we have that  b

>0

p=1

b

K(x, s)Φ(x)Φ(s)dσ(x)dσ(s) = a

 ηp2

a

m m  ηp2 1  2 < η < 0. λ(p) λ(1) p=1 p p=1

Let us now use the fact that the system of points si , which was discussed above, can always be chosen sufficiently dense to make all the differences m 

K(sj , sk )ϕ(p) (sj )ϕ(q) (sk )Δσj Δσk −

j,k=1

(Δσj = σ(sj ) − σ(sj−1 );

δpq λ(p)

(p, q = 1, 2, . . . , m)

j = 1, 2, . . . , n;

s0 = a)

arbitrarily small in absolute values. This will allow us to state that for a sufficiently dense system of points s1 < s2 < · · · < sn the quadratic form of the variables η1 , η2 , . . . , ηm n 

K(sj , sk )Φ(sj )Φ(sk )Δσj Δσk =

m 

apq ηp ηq

p,q=1

j,k=1

is negative definite and thus can be expanded in a sum of m negative squares. On the other hand, this form is obtained from the form F by the linear substitution ξj = Δσj

m 

ϕ(p) (sj )ηp

(j = 1, 2, . . . , m),

p=1

and for any linear substitution the number of negative squares in a form (as with the number of positive ones) cannot increase37 37 Indeed,

if in a form n  j,k=1

2 ajk ξj ξk = −X12 − X22 − · · · − Xν2 + Xν+1 + · · · + Xr2

242

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

Thus, m ≤ ν + , and consequently, as the set (193) is arbitrary, the integral equation (190) has a finite number N ≤ ν + of negative eigenvalues. Since by assumption the function σ(x) in (190) is strictly increasing, we have the expansion (see Section 1, (15)): K(x, s) =

∞  ϕj (x)ϕj (s) λj j=0

(a ≤ x, s ≤ b),

in which we can assume λ0 < λ1 < · · · < λN −1 < 0 < λN < . . . But then F =

∞ N −1 n    Zk2 Zk2 = + Aij ξi ξj , λk λk i,j=1

k=0

k=0

where n  i,j=1

Aij ξi ξj =

∞  Zk2 , λk

k=N

Zk =

n 

ϕk (sj )ξj

(k = 0, 1, 2, . . . ).

j=1

Thus F is obtained from a non-negative form by subtracting N squares and therefore, by Theorem 17 of Chap. I, it has at most N negative eigenvalues; i.e., ν + ≤ N . This proves the theorem. Remark I. For the integral equation (190) which satisfies the conditions of Theorem 17, we can generalize Theorem 13 of Section 9 by showing that for all λ less than the smallest eigenvalue λ0 of this equation, its resolvent Γ(x, s; λ) is an oscillatory single-pair kernel. Since for every real regular value γ the kernel Γ(x, s; −γ) has the same fundamental functions as the initial kernel K(x, s), and the eigenvalues of these kernels, λj and λj (j = 0, 1, 2, . . . ), are connected by the relations λj = λj + γ (j = 0, 1, 2, . . . ), the successive fundamental functions of (190), which satisfy the conditions of Theorem 17, have the whole set oscillatory properties. If Γ1 (x, s; λ) is the resolvent corresponding to the kernel K1 (s, x) = Γ(x, s; −γ) (for the same mass distribution function σ(x)), then Γ1 (x, s; λ) = Γ(x, s; λ−γ) and, in particular, K(x, s) = Γ1 (x, s; γ). Using this fact, and also Theorems 14 and 15 of Section 9, one can prove that for a single-pair kernel K(x, s), the conditions 1◦ and 2◦ of Theorem 17 are necessary and sufficient for the everywhere loaded integral equation (190) to have a resolvent Γ(x, s; λ) which is an oscillatory kernel for at least one value of λ (and therefore for all values which are less than the smallest eigenvalue of (190)). having ν negative squares, one makes a linear transformation from the variables ξj (j = 1, 2, . . . , n) n); then the linear functions Xi (i = 1, 2, . . . , r) will be to the variables ηp (p = 1, 2, . . . , m; m ≥ ≤ transformed into linear functions Yi (i = 1, 2, . . . , r) (which may well be linearly dependent) and r  a new form will in any case be obtained from the non-negative form Yi2 by subtracting ν ν+1

squares Yi2 (i = 1, 2, . . . , ν). Consequently, by Theorem 17 of Chap. I, it will have at most ν negative eigenvalues.

10. THE STURM–LIOUVILLE EQUATIONS

243

Remark II. If in an integral equation (190) the kernel K(x, s) is the Green function of the boundary value problem (183) and dσ(x) = r(x)dx, then in the representation (164) of Section 9 of the resolvent Γ(x, s; λ), the functions ψ(x; λ) and χ(x; λ) are determined for any λ as solutions to the differential equation L[ϕ] − λr(x)ϕ = 0 with initial conditions ϕ(a) = ψ(a),

  dϕ  dψ  p =p dx x=a dx x=a

ϕ(b) = χ(b),

  dχ  dϕ  =p p dx x=b dx x=b

for ψ(x; λ) and

for χ(x; λ). This readily follows from (158) and (160) of Section 9, which define ψ(x; λ) and χ(x; λ), if it is taken into account that in the case under consideration the function V (x, s) is the Cauchy function of the operator L[ϕ]. This means that for every integrable function f (x) the integral   x  b V (x, s)f (s)ds respectively V (s, x)f (s)ds a

x

gives a solution to the equation L[ϕ] = f with zero initial values of ϕ and p dϕ/dx at the point a (respectively b). As a consequence of this, Theorems 14 and 15 of Section 9 for this case of integral equation (190) give the well-known Sturm oscillation theorems concerning solutions to differential equations of second order in the presence of a parameter λ, which enters linearly. To conclude this section, we notice that although we have considered the Sturm–Liouville boundary value problem in rather general formulation (usually the problem is considered under more restrictive conditions with regard the coefficients of the equation p(x), q(x), and r(x)), we have not exhausted all potential possibilities opening as a result of our integro-algebraic investigations. Indeed, if we take for example the integral equation (190) with Green function K(x, s) of the operator L[ϕ], it will have a meaning for any monotone distribution function σ(s) (without requiring that it be absolutely continuous), and a theory of such equations exists. If we begin with such an arbitrary distribution function σ(x), then the integral equation (190) will be equivalent not to a system (183), but to a properly interpreted integro-differential system38 : x  x  x ⎫ dϕ  ⎪ − q(x)ϕ(x)dx + λ ϕ(x)dσ(x) = 0,⎪ p ⎪ ⎪ dx a ⎪ a a ⎪ ⎪   ⎬  dϕ  (194) sin α · ϕ + cos α · p = 0,  ⎪ dx x=a ⎪ ⎪   ⎪ ⎪ ⎪ dϕ  ⎪ ⎭ sin β · ϕ + cos β · p = 0. dx x=b 38 By

f |x a we denote f (x) − f (a).

244

IV. SYSTEMS WITH INFINITE NUMBER OF DEGREES OF FREEDOM

If, in addition, we recall the general interpretation of the concept of elastic support of a string (see Section 7) then the next natural step of generalization will be to replace the integro-differential equation in (194) by the following equation: x  x  x dϕ  p − ϕ(x)dτ (x) + λ ϕ(x)dσ(x) = 0, dx a a a where τ (x) is an arbitrary function of bounded variation. The statement at the end of remark I allows us to state that the fundamental functions of this general boundary value problem will also have a full set of oscillatory properties, provided σ(x) is strictly increasing.

http://dx.doi.org/10.1090/chel/345/06

CHAPTER V

Sign-Definite Matrices The present chapter contains several generalizations and supplements to the theory of oscillatory matrices which was developed in Chap. II. 1. Basic definitions Definition 1. A rectangular matrix A = aik 

(i = 1, 2, . . . , m; k = 1, 2, . . . , n)

will be called sign-definite of class d (d ≤ m, n), if for every p ≤ d, all non-vanishing minors of order p have the same sign, εp . If in addition, for every p ≤ d, all minors of order p are different from 0, we shall call the matrix A strictly sign-definite. In particular, when ε1 = ε2 = · · · = εd = 1, sign-definite (or strictly signdefinite) matrix of class d is called totally non-negative (or respectively totally positive) of class d. The notion of an oscillatory matrix is generalized in the following manner. Definition 2. A square sign-definite matrix A = aik n1 of class d (d ≤ n) will be called a matrix of class d+ if some power of A is strictly sign-definite of class d. Obviously, an oscillatory matrix A = aik n1 is totally non-negative of class n+ . Remark. In the case when d = min(m, n) we omit the words “of class d” and speak merely of sign-definite, strictly sign-definite, totally non-negative and totally positive matrices. Let us give examples of sign-definite matrices. Example 1. Any totally non-negative (totally positive) matrix is sign-definite (respectively strictly sign-definite). Example 2. If in any totally non-negative (or totally positive) matrix A we renumber all the rows (or all the columns) in reverse order, we obtain a sign-definite (or respectively strictly sign-definite) matrix. In this case for every p ≤ m, n we have εp = (−1)p(p−1)/2 . Example 3. A single-pair matrix L = lik n1 with elements  ψi χk (i ≤ k) (i, k = 1, 2, . . . , n), lik = ψk χi (i ≥ k) where all numbers ψ1 , . . . , ψn , χ1 , . . . , χn are different from 0, is sign-definite if and only if the numbers ψ1 , ψ2 , . . . , ψn have the same sign εψ , the numbers χ1 , χ2 , . . . , χn 245

246

V. SIGN-DEFINITE MATRICES

have the same sign, εχ , and one of the following two systems of inequalities is satisfied: ψ1 ψ2 ψn a) ≤ ≤ ... , χ1 χ2 χn or ψ1 ψ2 ψn b) ≥ ≥ ... . χ1 χ2 χn In case a) εp = εψ εχ , in case b) εp = (−1)p−1 εψ εχ . The rank r of L equals the number of < signs (or respectively the number of > signs) increased by one, in the inequalities a) or b). These statements follow from the expressions for the minors of arbitrary order of a single-pair matrix in terms of the numbers ψ1 , . . . , ψn , χ1 , . . . , χn (see Example 5 of Section 3, Chap. II). In Section 5 we shall prove the existence of sign-definite matrices of class d with any prescribed distribution of signs ε1 , ε2 , . . . , εd of the minors. We note the simplest properties of sign-definite matrices of class d. 1◦ The product of two sign-definite (or strictly sign-definite) matrices of class d is also a sign-definite (strictly sign-definite) matrix of class d. 2◦ If A = aik n1 is a sign-definite (strictly sign-definite) matrix of class d, then 2 A is totally non-negative (totally positive) matrix of class d. 3◦ If A = aik n1 is a sign-definite matrix of class d, and some power Al is strictly sign-definite of class d, then for every m > l the matrix Am is strictly sign-definite of class d. It will be proved in Section 4 that if A = aik n1 is a sign-definite matrix of class + d , then the first d eigenvalues of largest moduli and the corresponding eigenvectors of the matrix A have the same properties as the eigenvalues and eigenvectors of an oscillatory matrix (see Theorem 6 of Section 5, Chap. II). 2. Oscillating systems of vectors We introduce the following definition: Definition 3. A system of vectors ui = (u1i , u2i , . . . , uni ) (i = 1, 2, . . . , m) m ( c2i > 0) the vector has the property (T + ) (or T − ) if for every c1 , c2 , . . . , cm 1

u = c1 u + c2 u + · · · + cm u 1

2

m

satisfies the inequality1 Su+ ≤ m − 1 (respectively

Su− ≤ m − 1).

Theorem 1. In order for a system of vectors uk = (u1k , u2k , . . . , unk ) (k = 1, 2, . . . , m; m ≤ n) to have property (T + ), it is necessary and sufficient that all minors   i1 i2 . . . im (1 ≤ i1 < i2 < · · · < im ≤ n) U (1) 1 2 ... m be different from 0 and have the same sign. 1 The

− and S + were introduced in the beginning of Sec. 5, Chap. II. symbols Su u

2. OSCILLATING SYSTEMS OF VECTORS

247

Proof. We first prove the necessity of the conditions. We note first of all that if a minor (1) vanishes, one can find numbers ck (k = 1, 2, . . . , m), not all vanishing, from the system of equations m 

c k ui ν k = 0

(ν = 1, 2, . . . , m).

k=1

Then if u = (u1 , u2 , . . . , un ) is defined by u =

m 

ck uk , we have

1

ui1 = ui2 = · · · = uim = 0, and consequently Su+ ≥ m. Thus, all minors (1) are different from 0. In order to prove that all minors (1) have the same sign, it is enough to show that the m + 1 minors of the m-th order generated by any m + 1 rows of the matrix uik 

(i = 1, 2, . . . , n;

k = 1, 2, . . . , m)

have the same sign. Without loss of generality we can assume that the minors of interest to us are   1 ... i − 1 i + 1 ... m+ 1 (i = 1, 2, . . . , m + 1), Ui = U 1. . . . . . . . . . . . . . . . . . . . . . . . m since any other case will differ from this only by a different enumeration of the rows. In order to show that any two of these minors Ug and Uh (g < h) have the same sign, we define the numbers u1,m+1 , u2,m+1 , . . . , um+1,m+1 by ⎧ g−1 ⎪ ⎨ (−1) Uh for i = g, ui,m+1 = (−1)h Ug for i = h, (2) ⎪ ⎩ 0 for i = h, g. Then

 U

1 ... 1 ...

m+1 m+1

 =

m+1 

(−1)m+i+1 ui,m+1 Ui = 0.

i=1

We can therefore define numbers c1 , c2 , . . . , cm+1 , not all 0, which satisfy the system of homogeneous equations m+1 

ck uik = 0

(i = 1, 2, . . . , m + 1).

(3)

k=1

Let us consider now the vector u = c1 u1 + c2 u2 + · · · + cm um ; we denote its coordinates by u1 , u2 , . . . , un . Equations (2) and (3) show that ⎫ u1 = · · · = ug−1 = 0;⎪ ⎬ ug+1 = · · · = uh−1 = 0; ug = −cm+1 ug,m+1 = (−1)g cm+1 Uh ; (4) ⎪ ⎭ h+1 cm+1 Ug ; uh+1 = · · · = um+1 = 0. uh = −cm+1 uh,m+1 = (−1) It follows from this that all numbers c1 , . . . , cm cannot simultaneously vanish, for then we would have ug = 0, and thus cm+1 = 0. Therefore, by the condition of the theorem, Su+ ≤ m − 1. Next, cm+1 = 0, for otherwise the m + 1 coordinates of the vector u would be 0, and then Su+ ≥ m. But in this case it follows from the

248

V. SIGN-DEFINITE MATRICES

same equation (4) that Ug and Uh have the same sign; for otherwise, by assigning to each zero ui the sign (−1)i sign (cm+1 Uh ), we would have m + 1 sign changes among the m + 1 coordinates of u. Thus, the necessity of the condition of the theorem is proved. Let us proceed now to prove the sufficiency. We assume that the condition of the theorem is not sufficient; i.e., that all minors (1) are different from 0 and m  have the same sign, while at the same time for the vector u = ck uk with some k=1 m 2 + ck (k = 1, 2, . . . , m; k=1 ck > 0) we have Su ≥ m. We can then choose m + 1 coordinates of the vector u: ui1 , ui2 , . . . , uim+1 , such that (λ = 1, 2, . . . , m). (5) uiλ uiλ+1 ≤ 0 Note that ui1 , ui2 , . . . , uim cannot be all simultaneously equal to 0, for otherwise the numbers ck (k = 1, 2, . . . , m) would satisfy the system of homogeneous equations m 

c k ui λ k = 0

(λ = 1, 2, . . . , m)

k=1

with non-vanishing determinant (1). Let us now consider the following determinant, which is equal to 0:   ... ui 1 m ui 1   ui1 1   ... ui 2 m ui 2   ui 2 1    . . . . . . . . . . . . . . . . . . . . . . . . . . . .  = 0.    ............................    uim+1 1 . . . uim+1 m uim+1 Expanding by the last column, we obtain  m+1  i1 . . . iλ−1 iλ+1 . . . m+λ+1 (−1) ui λ U 1 ....................... λ=1

im+1 m

 = 0.

But such an equality is impossible, for by virtue of (5) and the condition of the theorem, all non-vanishing terms in the left hand side (and there are such terms) are of the same sign. This proves the theorem. Definition 4. A system of vectors ak = (a1k , a2k , . . . , ank )  (k = 1, 2, . . . , m) m has property (D+ ) (or property (D− )) if for every c1 , c2 , . . . , cm ( k=1 c2k > 0) the vector a = c 1 a 1 + c2 a 2 + · · · + cm a m satisfies the inequality2 Sa+ ≤ Sc−

2 Here

(respectively Sa− ≤ Sc− ).

Sc− denotes the minimum number of sign changes in the sequence c1 , c2 , . . . , cm .

2. OSCILLATING SYSTEMS OF VECTORS

249

Theorem 2. In order for a system of vectors ak = (a1k , a2k , . . . , ank ) (k = 1, 2, . . . , m; m < n) to have property (D+ ), it is necessary and sufficient that the matrix A = aik  (i = 1, 2, . . . , n; k = 1, 2, . . . , m) be strictly sign-definite. Proof. Necessity of the condition. Let us choose an arbitrary system of indices 1 ≤ k1 < k2 < · · · < kp ≤ m, 1 ≤ p ≤ m, put all ck = 0 for k different from each kν (ν = 1, 2, . . . , p), and set a=

m 

ck ak =

p 

ck ν ak ν .

ν=1

k=1

Sc−

≤ p − 1, it follows that Sa+ ≤ p − 1. Therefore the vectors Since we have k1 kp a , . . . , a have the property (T + ), and by the preceding theorem, all minors   i1 i2 . . . ip (1 ≤ i1 < i2 < · · · < ip ≤ n) A k1 k2 . . . kp for this choice of k1 , k2 , . . . , kp , are different from 0 and have the same sign, which we denote by ε(k1 , k2 , . . . , kp ). It remains to show that the sign ε(k1 , . . . , kp ) is independent of the choice of k1 , k2 , . . . , kp and depends only on p. When p = m there exists only one system of indices k1 < k2 < · · · < km , namely, k1 = 1, k2 = 2, . . . , km = m. It remains therefore to consider only the case, that p < m. Let us take arbitrary indices 1 ≤ k1 < k2 < · · · < kp < kp+1 ≤ m. We shall show that for every ν = 1, 2, . . . , p ε(k1 , . . . , kν−1 , kν+1 , . . . , kp+1 ) = ε(k1 , . . . , kν , kν+2 , . . . , kp ).

(6)

For this purpose we consider a system of p vectors = aν−1 , aν∗ = dν aν + dν+1 aν+1 , a1∗ = a1 , . . . , aν−1 ∗ = aν+2 , . . . , ap+1 = ap+1 , aν+2 ∗ ∗ where dν and dν+1 are arbitrary positive numbers. If we set a=

p+1  i=1 (i=ν+1)

c∗i ai∗

=

p+1 

ck ak ,

k=1

then evidently

cν = c∗ν dν , cν+1 = c∗ν dν+1 , and so cν cν+1 ≥ 0. Consequently, Sc− ≤ p − 1 and Sa+ ≤ Sc− ≤ p − 1. Thus, the , . . . , ap+1 have property (T + ) and therefore, according to vectors a1∗ , . . . , aν∗ , aν+2 ∗ ∗ Theorem 1, all minors     i . . . . . . . . . . . . . . . ip i . . . . . . . . . . . . . . . ip = dν A 1 A∗ 1 1, . . . , ν, ν + 2, . . . , p + 1 1, . . . , ν, ν + 2, . . . , p + 1   (7) i1 . . . . . . . . . . . . . . . ip +dν+1 A 1, . . . , ν − 1, ν + 1, . . . , p + 1 are different from 0 (A∗ = a∗ik , ak∗ = (a∗1k , . . . , a∗nk ), i = 1, 2, . . . , n; k = 1, . . . , ν, ν + 2, . . . , p + 1). Were equation (6) not to hold, we could choose positive numbers dν and dν+1 such that one of the minors (7) would vanish, but this is impossible.

250

V. SIGN-DEFINITE MATRICES

Thus (6) holds, i.e., ε(k2 , . . . , kp+1 ) = ε(k1 , k3 , . . . , kp+1 ) = · · · = ε(k1 , . . . , kp ). From this we conclude that ε(k1 , k2 , . . . , kp ) will not change if we change one index, and hence it is independent of the choice of k1 , k2 , . . . , kp . Sufficiency of the condition. Let the matrix A be strictly sign-definite and let Sc− = p. We assume, for definiteness, that the first non-vanishing coordinate of c is positive. Then the coordinates of c are split into p + 1 groups in the following manner: c1 ≥ 0, c2 ≥ 0, . . . , cν1 ≥ 0; cν1 +1 < 0, cν1 +2 ≤ 0, . . . , cν2 ≤ 0; ν1  ck > 0). cν2 +1 > 0, cν2 +2 ≥ 0 etc. ( k=1

If we put u1 =

ν1 

|ck |ak ,

k=1

then the vector a =

m 

u2 =

ν2 

|ck |ak

etc.,

k=ν1 +1

ck ak can be represented as

k=1

a = u1 − u2 + u3 − · · · + (−1)p up+1 . On the other hand, using the theorem on the addition of determinants, we can readily verify that the minors of order (p + 1) of the matrix U = uik  (k = 1, 2, . . . , p + 1; i = 1, 2, . . . , n; uk = (u1k , . . . , unk )) are different from 0 and have the same sign as the minors of order (p + 1) of the matrix A. Therefore, by Theorem 1, Sa+ ≤ p = Sc− . This proves the theorem. It is obvious that this theorem can be restated as follows: Theorem 2 . Assume that a linear transformation yi =

m 

aik xk

(i = 1, 2, . . . , n; m < n)

k=1

is given. Then in order for every vector x = (x1 , x2 , . . . , xm ) ( the inequality Sy+ ≤ Sx− ,



x2i > 0) to satisfy (8)

it is necessary and sufficient that the matrix A = aik  be strictly sign-definite. Remark. When m ≥ n the strict sign-definiteness of A is not necessary for the system of vectors ak (k = 1, 2, . . . , m) to have property (D+ ), or (what is the same) for (8) to be satisfied. Indeed, consider the transformation y1 = 2x1 + 2x2 , y2 = x1 + x2 .

2. OSCILLATING SYSTEMS OF VECTORS

251

Here m = n = 2. For this transformation and for x21 + x22 > 0 we always have Sy+ ≤ Sx− ; at the same time the matrix   2 2   1 1 is not strictly sign-definite. Let us also consider the transformation (m = 3, n = 2): y1 = 2x1 + 3x2 + 2x3 , y2 = x1 + 3x2 + x3 . For this transformation we always have Sy+ ≤ Sx− . Indeed, let Sy+ = 1, for example, y1 ≥ 0 and y2 ≤ 0. Then y1 −y2 = x1 +x3 ≥ 0, and therefore 3x2 = y2 −(x1 +x3 ) ≤ 0. Let us consider the first case: x2 < 0. Then x 1 + x3 =

1 3 y1 − x2 > 0, 2 2

and, consequently either x1 > 0 or x3 > 0; thus Sx− = 1 or 2. Let us consider another case: x2 = 0. Then x1 + x3 = y2 + y1 /2, and consequently (since y1 ≥ 0 and y2 ≤ 0) x1 + x3 = 0. Therefore if not all the xi = 0 (i = 1, 2, 3), then x1 x3 < 0 and Sx− = 1. However, the matrix   2 3 2   1 3 1, is evidently not sign-definite. When m ≥ 2, Theorems 2 and 2 remain in force if the condition “the matrix A is strictly sign-definite” is replaced by the condition “the matrix A is strictly sign-definite of class n − 1.” The sufficiency of this condition follows from the fact that Sc− ≥ n − 1 implies Sa+ ≤ Sc− , since the vector a has n coordinates and therefore Sa+ ≤ n − 1, and when Sc− < n − 1, all the arguments in the proof of Theorem 2 remain in force. The necessity of the condition is proved exactly the same way as in the proof of Theorem 2. Theorem 3. In order for a system of linearly independent vectors uk = (u1k , u2k , . . . , unk )

(k = 1, 2, . . . , m)

to have the property (T − )3 , it is necessary and sufficient that all non-vanishing minors   i1 i2 . . . im (1 ≤ i1 < i2 < · · · < im ≤ n) U 1 2 ... m be of the same sign 4 . 3 See

Definition 3 in Section 2. this theorem follows directly Lemma 4 of Section 5, Chap. II.

4 From

252

V. SIGN-DEFINITE MATRICES

Proof. Necessity of the condition. We introduce the totally positive 2 matrix Fσ = e−σ(i−k) n1 (σ > 0) (see Example 2, Section 3, Chap. II), and construct the vectors ui (σ) = Fσ ui

(i = 1, 2, . . . , m).

Since Fσ → E as σ → +∞, then ui (σ) → ui

as σ → +∞

Let us consider the vector u =

m 

(i = 1, 2, . . . , m).

ci ui . Then

i=1

u(σ) = Fσ u = Fσ (

m 

c i ui ) =

i=1

m 

ci ui (σ).

i=1

Su−

+ ≤ m−1, then it follows from Theorem 2 that Su(σ) ≤ m−1; Since by assumption i.e., the maximum number of sign changes in the vector

u(σ) =

m 

ci ui (σ)

i=1

does not exceed m − 1. Consequently, by Theorem 1, in the matrix Uσ = uik (σ) (i = 1, 2, . . . , n; k = 1, 2, . . . , m) all minors   i i . . . im (1 ≤ i1 < i2 < · · · < im ≤ n) Uσ 1 2 (9) 1 2 ... m are of the same sign. Since    i1 i i . . . im →U Uσ 1 2 1 2 ... m 1

i2 2

... ...

im m

 as

σ → +∞,

we see that the condition of the theorem is necessary. The condition is sufficient. Introducing again the vectors ui (σ) = Fσ ui (i = 1, 2, . . . , m) and using the Binet–Cauchy theorem on the product of two rectangular matrices, we obtain   i1 i2 . . . im Uσ 1 2 ... m      i1 i2 . . . im α1 α2 . . . αm . U = Fσ α1 α2 . . . αm 1 2 ... m 1≤α1 0 , k=1

then we can set ⎧ ⎪ ⎪ ⎨ bij = ⎪ ⎪ ⎩

νj 

aik |xk |

(j = 1, . . . , p; ν0 = 0; i = 1, 2, . . . , p),

k=νj−1 +1

ξj = (−1)j−1

(j = 1, 2, . . . , p)

so that yj =

p 

bij ξj

(i = 1, 2, . . . , p).

j=1

As the matrix B = bij  (i = 1, 2, . . . , n; j = 1, 2, . . . , p) is totally positive and Sy− = Sξ = p − 1, we are back to the case previously considered, and consequently the vector y oscillates in the same way as ξ and so in the same way as x. Suppose now that the matrix A is not totally positive, but totally non-negative. Assume that (15) holds for some x = 0. Let us introduce the totally positive matrix 2 Fσ = e−σ(i−k) n1 (σ > 0) and define the vector y(σ) = Fσ y = Fσ Ax = Aσ x. Since y(σ) → y as σ → ∞, then for σ large enough − Sy(σ) ≥ Sy− = Sx− .

Since the matrix Aσ = Fσ A is totally positive5 , it follows from Theorem 2 that + ≤ Sy(σ) ≤ Sx− , and consequently,

− Sy(σ)

− Sy(σ) = Sx− ;

thus, according to the previous proof, y(σ) oscillates in the same way as x. On the other hand, the vector y(σ) oscillates in the same way as y. Consequently, y oscillates in the same way as x. This proves the theorem. Theorem 6. Let there be given a system of n independent homogeneous equations with m variables (m > n) ⎫ a11 x1 + a12 x2 + · · · + a1m xm = 0, ⎪ ⎪ ⎪ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .⎬ (16) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .⎪ ⎪ ⎪ ⎭ an1 x1 + an2 x2 + · · · + anm xm = 0. 5 Because

the matrix Fσ is totally positive, and A is totally non-negative and has rank m.

2. OSCILLATING SYSTEMS OF VECTORS

257

In order that every non-trivial solution x = (x1 , x2 , . . . , xm ) of this system have the property Sx− ≥ n, it is necessary and sufficient that all minors of order n of the matrix A = aik  be different from 0 and of the same sign. Proof. The condition is necessary. Indeed, if for some 1 ≤ k1 < k2 < · · · < kn ≤ m   1 2 ... n A = 0, (17) k1 k2 . . . kn there would be a non-trivial solution of the system (16) in which all xi = 0 when i is different from k1 , k2 , . . . , kn ; but this would mean that Sx− < n. Thus equation (17) is impossible. To prove that all minors   1 2 ... n A (1 ≤ k1 < k2 < · · · < kn ≤ m) k1 k2 . . . kn have the same sign, it is enough to show that for every 1 ≤ k1 < k2 < · · · < kn+1 ≤ m all minors   1 2 ... ... ... n Δν = A k1 . . . kν−1 kν+1 . . . kn+1 are of the same sign. To do this, we notice that by putting xi = 0 for i different from k1 , . . . , kn+1 and xkν = (−1)ν Δν (ν = 1, 2, . . . , n + 1), we obtain a solution of the system (16). Since Sx− = n for this solution, we conclude that all the Δν (ν = 1, 2, . . . , n) are of the same sign. The condition is sufficient. When this condition is satisfied, at least n + 1 of the coordinates of any solution x = (x1 , x2 , . . . , xm ) = 0 of (16) differ from 0, for in the opposite case, by discarding the m − n numbers xi which are zero, we would have for the remaining n numbers xk1 , xk2 , . . . , xkn the system of equations n 

aikν xkν = 0

(i = 1, 2, . . . , n),

ν=1

which has only zero as solution. If now Sx− < n, we divide the numbers xi (i = 1, 2, . . . , m) into n groups: x1 , x2 , . . . , xν1 ; xν1 +1 , . . . , xν2 ; . . . ; xνn−1 +1 , . . . , xνn so that each group has the same sign: xk = εj |xk |, νj−1 < k ≤ νj

(j = 1, 2, . . . , n; ν0 = 0),

and each group contains at least one non-zero number. We put bij =

νj 

aik |xk |

(i, j = 1, 2, . . . , n).

k=νj−1 +1

Then

n  j=1

bij εj = 0

(i = 1, 2, . . . , n).

258

V. SIGN-DEFINITE MATRICES

On the other hand,   1 2 ... n = B 1 2 ... n



 |xk1 xk2 . . . xkn |A

1≤k1 ≤ν1 ν1 n) m 

aik xk = 0

(i = 1, 2, . . . , n).

(18)

k=1

In order for every non-trivial solution x = (x1 , x2 , . . . , xm ) to have the property Sx+ ≥ n, it is necessary and sufficient that all non-vanishing minors of order n of the matrix A = aik  have the same sign. Proof. We introduce the matrix Fσ = e−σ(i−k) m 1 2

ξ(σ) =

Fσ−1 x,

or

(σ > 0). We put

x = Fσ ξ(σ).

Substituting for each xi in (18) its expression in terms of ξk , we obtain the system m 

aik (σ)ξk = 0

(i = 1, 2, . . . , n),

(19)

k=1

where Aσ = aik (σ) = AFσ . If Sx+ ≥ n for every non-zero solution x of the system (18), then we conclude by − Theorem 2 that Sξ(σ) ≥ n for every non-zero solution ξ(σ) of the system (19), and consequently, by the preceding theorem, all minors of order n of the matrix Aσ are different from zero and are of the same sign. Passing to the limit as σ → +∞, we obtain the necessity of the condition of the theorem. In the other direction, suppose that the condition of the theorem is satisfied. Then all minors of n-th order of Aσ are different from zero and have the same sign; − therefore, in accordance with the preceding theorem, Sξ(σ) ≥ n. Passing to the + limit as σ → +∞, we obtain Sx ≥ n. This proves the theorem. 3. Markov systems of vectors 1. We introduce the notion of a Markov system of vectors which is analogous to the notion of a Markov sequence of functions (see Definition 4, Sect. 4, Chap. IV). Definition 6. Vectors u 1 , u 2 , . . . , up

(p ≤ n)

form a Markov system if for every k ≤ p the vectors u1 , u2 , . . . , uk have the property (T + ). From this definition and Theorem 1 we obtain the

3. MARKOV SYSTEMS OF VECTORS

259

Corollary. In order for a system of vectors ( j % u1 , u2 , . . . , up u = (u1j , u2j , . . . , unj ); j = 1, 2, . . . , p to be a Markov system, it is necessary and sufficient that for every k ≤ p all determinants   i1 i2 . . . ik (1 ≤ i1 < i2 < · · · < ik ≤ n) U 1 2 ... k be different from zero and have the same sign εk ; i.e.,    2  εk = 1; 1 ≤ i1 < i2 < · · · < ik ≤ n i1 i2 . . . ik . >0 εk U k = 1, 2, . . . , p 1 2 ... k

(20)

It turns out that not all inequalities (20) are independent. Fekete [13] has shown that it is possible to confine oneself only to those inequalities of (20) which contain the minors formed by adjacent rows and columns. We proceed to a clarification of this point. 2. Definition 7. By density of a minor     i1 . . . ip i1 < i2 < · · · < ip U k1 . . . kp k1 < k2 < · · · < kp of the matrix uik n1 we understand the number μ defined by μ=

p−1 

(iν+1 − iν − 1) +

ν=1

p−1 

(kν+1 − kν − 1) =

ν=1

= ip − i1 + kp − k1 − 2(p − 1). According to this definition, the minors of zero density are the minors formed by adjacent rows and adjacent columns:   i i + 1 ... i + p − 1 . U k k + 1 ... k + p − 1 In the following lemma we establish an identity which expresses the minors of a given density in terms of minors of a lower density. Lemma 1. For an arbitrary matrix uij  with k + 1 rows and k columns, the following equality holds     2 ... k − 1 k + 1 1 2 ... k − 1 k U U 1 ... . k−1 1 . ... . k     2 ... k − 1 1 k 2 ... k − 1 k + 1 (21) U +U 1 ... . k−1 1 . ... . k     2 ... k − 1 k k + 1 2 ... k − 1 1 = 0. U +U 1 ... . k−1 1 . ... . k

260

V. SIGN-DEFINITE MATRICES

Proof. Consider the determinant   ... u1k u11 ... u1,k−1   u11    ............................................     uk+1,1 . . . uk+1,k uk+1,1 . . . uk+1,k−1   . ... u2k 0 ... 0  u21     ............................................    uk−1,1 . . . uk−1,k 0 ... 0 This determinant vanishes, since all minors of (k + 1)-th order contained in the first k + 1 rows vanish. Writing the Laplace expansion for this determinant in terms of the minors of the first k columns, we obtain (21). Let there be given a matrix U = uij 

(i = 1, 2, . . . , n; j = 1, 2, . . . , p; p ≤ n).

Choose an arbitrary minor of k-th order,   i1 i2 . . . ik U 1 2 ... k

(22)

(k ≤ p)

with density μ > 0. Then for some h < k ih+1 − ih ≥ 2, and consequently we can choose an index i so that ih < i < ih+1 . We now use (21), replacing in it the rows 1, 2, . . . , k − 1, k, k + 1 by the rows i1 , i2 , . . . , ik−1 , i, ik . We then obtain     i2 . . . ik−1 ik i1 . . . ik−1 i U U 1 ... . k 1 ... . k−1     i2 . . . ik−1 i1 i i2 . . . ik−1 ik U +U 1 . ... . k 1 ... . k−1     i2 . . . ik−1 i ik i2 . . . ik−1 i1 = 0; U +U 1 ... . k−1 1 . ... . k hence



 i1 . . . ik U 1 ... k      i2 . . . i1 . . . ih i ih+1 . . . ik−1 ik U = U 1 ... k 1 ... k − 1     i1 . . . ik−1 i2 . . . ih i ih+1 . . . ik U +U 1 ... k 1 ... k − 1 −1

 i2 . . . ih i ih+1 . . . ik−1 × U 1 ... k−1

This equation gives us an expression for the minors of k-th order   i1 i2 . . . ik (1 ≤ i1 < i2 < · · · < ik ≤ n; k = 1, 2, . . . , p) U 1 2 ... k

(23)

(24)

3. MARKOV SYSTEMS OF VECTORS

261

of (22) with density μ which is formed by the first k columns and arbitrary rows, in terms of similar minors of order k − 1 and minors of order k but having density < μ. It follows from (23) that if all minors (24) of order k − 1 and those of order k with density < μ are different from zero and have a sign ε, such that ε depends only on the order of the minor, then this property will also hold for all minors (24) of order k with density ≤ μ. Hence, by induction in k and μ, we arrive at the following theorem: Theorem 8 (Fekete).

In order that a system of vectors

uj = (u1j , u2j , . . . , unj )

(j = 1, 2, . . . , p),

(p ≤ n)

be a Markov system, or, which is the same, in order for the matrix    u11 . . . u1p     ..............  (p ≤ n)    un1 . . . unp  to have all minors  i1 i2 . . . U 1 2 ...

ik k

 (1 ≤ i1 < i2 < · · · < ik ≤ n; k = 1, 2, . . . , p)

(25)

non-zero and of sign εk which depends only on the order k of the minor, it is sufficient that this property be possessed by all the minors (25) of zero density, i.e., the minors   i i + 1 ... i + k − 1 (i = 1, 2, . . . , n − k + 1; k = 1, 2, . . . , p). U 1 2 ... k Corollary. Let A = aik n1 . If for every p < n we have   i i + 1 ... i + p − 1 >0 (i, k = 1, 2, . . . , n − p + 1), εp A k k + 1 ... k + p − 1 where εp (p = 1, 2, . . . , n) are certain signs, then A is strictly sign-definite. In particular, if ε1 = · · · = εn = 1, then A is totally positive. 3. Let us prove two lemmas on the extension of chains of vectors which are Markov systems. These lemmas will be used in the next section. Lemma 2. A Markov system of vectors uj = (u1j , u2j , . . . , unj )

(j = 1, 2, . . . , p),

where p < n, can always be extended; i.e., it is always possible to find a vector up+1 such that the system of vectors u1 , u2 , . . . , up , up+1 is again a Markov system. Proof. It is sufficient to prove the existence of a vector up+1 such that   i i + 1 ... i + p >0 (i = 1, 2, . . . , n − p). U 1 2 ... p + 1

262

V. SIGN-DEFINITE MATRICES

Denoting the coordinates of the this vector up+1 by x1 , x2 , . . . , xn , we can write these inequalities as follows:   ... uip xi   ui1    ui+1,1 . . . ui+1,p xi+1  (i = 1, 2, . . . , n − p).  >0  .........................    ui+p,1 . . . ui+p,p xi+p The determinants in the left hand sides of the last inequalities are linear forms in x1 , x2 , . . . , xn . These forms are linearly independent, since the form corresponding to a given index i contains the variable xi+p with coefficient   i i + 1 ... i + p − 1 = 0, U 1 2 ... p whereas the preceding forms (corresponding to smaller values of the index i) do not contain this variable. It follows from linear independence that it is possible to choose the values of x1 , x2 , . . . , xn , so that these forms assume any prescribed values and, in particular, positive values. This proves the lemma. Lemma 3. If p < n, every two biorthonormal Markov systems u1 , u2 , . . . , up and v 1 , v 2 , . . . , v p (uj = (u1j , . . . , unj ), v j = (v1j , . . . , vnj ), (ui , v j ) = δij (i, j = 1, . . . , p)) can always be extended. In other words, when p < n, one can find such vectors up+1 = (u1,p+1 , . . . , un,p+1 )

and

v p+1 = (v1,p+1 , . . . , vn,p+1 )

that the systems of vectors u1 , . . . , up , up+1 and v 1 , . . . , v p , v p+1 are biorthonormal Markov systems. Proof. According to the preceding lemma, there exist vectors up+1 and v∗p+1 ∗ such that the systems of vectors u1 , . . . , up , up+1 ∗

and

v 1 , . . . , v p , v∗p+1

are Markov systems. In order to make these systems biorthogonal, let us replace , v∗p+1 by the vectors up+1 ∗ up+1 = up+1 − ∗

p 

c i ui ,

v p+1 = v∗p+1 −

i=1

p 

di vi .

i=1

For arbitrary ci , di (i = 1, 2, . . . , p), the systems u1 , . . . , up+1 and v 1 , . . . , v p+1 are Markov systems. Let us put ci = (up+1 , v i ), di = (ui , v∗p+1 ). ∗ Then, as can be readily verified, the vector up+1 will be orthogonal to all the v i , while v p+1 will be orthogonal to the ui (i = 1, . . . , p): (up+1 , v i ) = (ui , v p+1 ) = 0 (i = 1, 2, . . . , p).

(26)

4. EIGENVALUES AND EIGENVECTORS OF SIGN-DEFINITE MATRICES

According to the Binet–Cauchy identity (see Section 1, Chapt. I)     i1 i2 . . . i1 i2 . . . ip+1 i j p+1 V |(u , v )|1 = U 1 2 ... p+ 1 1 2 ... i1 · · · > |λd | > |λd+1 | ≥ · · · ≥ |λn | ≥ 0,

(31)

then

εp (p = 1, 2, . . . , d; ε0 = 1). (32) εp−1 2◦ The eigenvectors uj = (u1j , u2j , . . . , unj ) (j = 1, 2, . . . , d) of the matrix A, which correspond to the d largest (in absolute value) eigenvalues λ1 , λ2 , . . . , λd have the property that for all integers r and s (1 ≤ r ≤ s ≤ d) and for arbitrary s  ci (i = r, r + 1, . . . , s; c2i > 0) the number of sign changes in the coordinates of sign λp =

the vector u =

s 

i=r

ci ui lies between r − 1 and s − 1:

i=r

r − 1 ≤ Su− ≤ Su+ ≤ s − 1.

(33)

In particular, there are exactly j − 1 changes of sign in the coordinates of the eigenvector uj (j = 1, 2, . . . , d): Suj = j − 1.

264

V. SIGN-DEFINITE MATRICES

3◦ The nodes of two successive eigenvectors uj and uj+1 (j = 1, 2, . . . , d − 1) alternate. We first prove the theorem for the case when A is a strictly sign-definite matrix of class d. Proof of 1◦ . Let us arrange the eigenvalues of the matrix A in the order of non-decreasing moduli: |λ1 | ≥ |λ2 | ≥ · · · ≥ |λn |. We consider the p-th associated matrix Ap (p ≤ d) (see Section 13, Chap. I). Obviously all elements of εp Ap are positive, and therefore the Perron theorem applies to this matrix. On the other hand, according to Kronecker’s theorem (Section 13, Chap. I) the eigenvalues of Ap are all possible products of p eigenvalues of A. The product of largest modulus will be λ1 , λ2 , . . . , λp . Thus, simultaneous application of Perron’s and Kronecker’s theorems yields that εp λ1 λ2 . . . λp > 0 (p = 1, 2, . . . , d)

(34)

and |λ1 λ2 . . . λp | > |λ1 λ2 . . . λp−1 λp+1 |

(p = 1, 2, . . . , d).

(35)

Comparing (34) with the inequality εp−1 λ1 λ2 . . . λp−1 > 0 (p = 1, 2, . . . , d), we obtain that

εp

εp−1 and from (35) we obtain

λp > 0 (p = 1, 2, . . . , d; ε0 = 1),

|λp | > |λp+1 |

(p = 1, 2, . . . , d).

From this follow (31) and (32). Proof of 2◦ . Let again p ≤ d. Corresponding to the largest (in modulus) eigenvalue λ1 λ2 . . . λp of the matrix εp Ap is the eigenvector with coordinates   i1 i2 . . . ip (1 ≤ i1 < i2 < · · · < ip ≤ n) (36) U 1 2 ... p (see Section 13, Chapt. I). Since all elements of the matrix εp Ap are positive when p ≤ d, we again apply the Perron theorem and conclude that when p ≤ d all determinants (36) are different from zero and have the same sign; i.e., the vectors u1 , u2 , . . . , ud form a Markov system, and, consequently, if   s s  u= c i ui c2i > 0; i ≤ r ≤ s ≤ d (37) i=r

i=r

then we have: Su+ ≤ s − 1. Let us assume that for this vector u we have Su− = q < r − 1. Let u = (u1 , u2 , . . . , un ). Without loss of generality, we can assume that u1 ≥ 0. Then the

4. EIGENVALUES AND EIGENVECTORS OF SIGN-DEFINITE MATRICES

265

coordinates of u can be partitioned into q + 1 successive groups such that in each group all coordinates have the same sign: u1 ≥ 0, . . . , uk1 ≥ 0; uk1 +1 ≤ 0, . . . , uk2 ≤ 0; . . . . . . ; (−1)q ukq +1 ≥ 0, . . . , (−1)q un ≥ 0.

(38)

Let us consider now the eigenvectors v j = (v1j , v2j , . . . , vnj ) (j = 1, 2, . . . , d) of the transpose matrix A which correspond to the eigenvalues λ1 , λ2 , . . . , λd . Since A is also a strictly sign-definite matrix of class d, then as we proved before, the vectors v 1 , v 2 , . . . , v p also form a Markov system, with (ui , v j ) = 0 (i = j; i, j = 1, 2, . . . , d). We define now a vector v = (v1 , v2 , . . . , vn ) by   i k1 k2 . . . kq (i = 1, 2, . . . , n). vi = V 1 2 ... q q + 1

(39)

(40)

It follows from (40) that v = f1 v 1 + f2 v 2 + · · · + fq+1 v q+1

(q + 1 < r),

(41)

 k1 k2 . . . kq = 0, 1 2 ... q since the vectors v 1 , v 2 , . . . , v q have property (T + ). Since the vectors v 1 , v 2 , . . . , v d form a Markov system, all determinants   h1 h2 . . . hq hq+1 (1 ≤ h1 < h2 < · · · < hq+1 ≤ n) V 1 2 ... q q + 1 

where

fq+1 = V

are different from zero and have the same sign. Therefore (see (40)), in the sequence of coordinates v1 , v2 , . . . , vn (42) only the coordinates vk1 , vk2 , . . . , vkq vanish and sign changes in the sequence (42) can occur only at these zeros. Therefore, by (38) we find that (u, v) =

n 

ui vi = 0,

(43)

i=1

since the sum

n 

ui vi does not contain two terms of different signs, and one of the

i=1

terms is known to be different from zero (the vector u has at least q + 1 coordinates that are different from zero). Inequality (43) contradicts (37), (39), and (41), and so Su− ≥ r − 1. This proves Proposition 2◦ . Proposition 3◦ follows from 2◦ . This was shown in the proof of Theorem 6 of Section 5, Chap. II. The corresponding arguments should be repeated verbatim. The theorem is proved for the case when A is a strictly sign-definite matrix of class d. Let now A be a sign-definite matrix of class d and let Aκ be strictly sign-definite of class d. Then (see 3◦ of Section 1) the matrix Aκ+1 will also be strictly signdefinite of class d. We again enumerate the eigenvalues of the matrix A in such a

266

V. SIGN-DEFINITE MATRICES

way that |λ1 | ≥ |λ2 | ≥ · · · ≥ |λn |. Applying our theorem to the matrices Aκ and , . . . , λκ+1 are real, distinct, non-zero, Aκ+1 , we conclude that λκ1 , . . . , λκd and λκ+1 1 d and |λκ1 | > |λκ2 | > · · · > |λκd | > |λκd+1 | ≥ · · · ≥ |λκn |. It follows that all the λ1 , λ2 , . . . , λd are real, different from zero, distinct, and |λ1 | > |λ2 | > · · · > |λd | > |λd+1 | ≥ · · · ≥ |λn |. Since the matrices A and Aκ have the same eigenvectors u1 , u2 , . . . , ud , corresponding to the first d largest (in modulus) eigenvalues, then since Propositions 2◦ and 3◦ hold for Aκ , they also hold for A. This proves the theorem completely. 2. We supplement somewhat our theorem on the oscillatory properties of signdefinite matrices of class d. Let A = aik n1 be a sign-definite matrix of class d+ . According to Theorem 9, the eigenvalues of A satisfy η1 λ1 > η2 λ2 > · · · > ηd λd > |λd+1 | ≥ · · · ≥ |λn | ≥ 0, where η1 , η2 , . . . , ηn are ±16 . Then each λj (j = 1, 2, . . . , d) is a simple real eigenvalue both for A and for its transpose A , and to each correspond eigenvectors (defined up to scalar factors) uj = (u1j , u2j , . . . , unj ) and v j = (v1j , v2j , . . . , vnj ) of A and A : Auj = λj uj , A v j = λj v j

(uj = 0, v j = 0;

j = 1, 2, . . . , d).

According to Theorem 9, the systems of vectors u1 , u2 , . . . , ud and v 1 , v 2 , . . . , v d are Markov systems. Therefore, for every p ≤ d, all minors   i1 i2 . . . ip (1 ≤ i1 < i2 < · · · < ip ≤ n) U 1 2 ... p are different from zero and have the same sign εp (p = 1, 2, . . . , d). In exactly the same way, all minors   i1 i2 . . . ip (1 ≤ i1 < i2 < · · · < ip ≤ n) V 1 2 ... p are different from zero and have the same sign εp (p = 1, 2, . . . , d). Multiplying the vectors u1 , u2 , . . . , ud and v 1 , v 2 , . . . , v d by ±1, we can arrange that εp = εp = 1 (p = 1, 2, . . . , d); i.e., we can achieve the following inequalities:     i1 i2 . . . ip i1 i2 . . . ip > 0, V >0 (44) U 1 2 ... p 1 2 ... p   1 ≤ i1 < i2 < · · · < ip ≤ n . p = 1, 2, . . . , d Next, since λj = λg for j = g, (45) 6η = ε ε j j j−1 (j = 1, 2, . . . , d; ε0 = 1), where εj is the sign of the minors of j-th order of the matrix A.

4. EIGENVALUES AND EIGENVECTORS OF SIGN-DEFINITE MATRICES

267

it follows that (uj , v g ) = 0 for On the other hand, according to the   i1 . . . U |(uj , v g )p1 = 1 ... i1 0

(p = 1, 2, . . . , d).

But then (uj , v j ) > 0 (j = 1, 2, . . . , d). Multiply the vectors uj by the positive factors cj =

1 (uj , v j )

(relations (44) and

(46) remain in force), and denote the resultant vectors again by uj (j = 1, 2, . . . , d); then (uj , v j ) = 1. (47) Equations (46) and (47) say that the systems u1 , u2 , . . . , ud , and v 1 , v 2 , . . . , v d are biorthonormal. We have thus proved the following theorem: Theorem 10. If A = aik n1 is a sign-definite matrix of class d with eigenvalues λ1 , . . . , λn (|λ1 | > |λ2 | > · · · > |λd | > |λd+1 | ≥ · · · ≥ |λn |), then we can choose eigenvectors uj and v j of A and A , corresponding to the eigenvalues λj (j = 1, 2, . . . , d), so that the systems of vectors u1 , . . . , ud

and

v1 , . . . , vd

are biorthonormal: (uj , v g ) = δjg (j, g = 1, 2, . . . , d) and the following inequalities hold:     i1 . . . ip i1 . . . ip > 0, V >0 (p = 1, 2, . . . , d). U 1 ... p 1 ... p Remark. Theorem 10 applies to an arbitrary oscillatory or strictly sign-definite matrix A = aik n1 (in these cases one must put d = n). 3. Let now A = aik n1 be a sign-definite matrix of class d+ and rank d. Then some power of A is a strictly sign-definite matrix of class d, and the zero eigenvalue of A has defect n − d. Consequently (since λ1 = 0, . . . , λd = 0; see Theorem 9) λd+1 = · · · = λn = 0. Corresponding to the eigenvalue 0 are n − d linearly independent eigenvectors ud+1 , . . . , un of A, and n − d linearly independent eigenvectors v d+1 , . . . , v n of A . We can obtain these vectors in the following manner.

268

V. SIGN-DEFINITE MATRICES

By (45), the eigenvectors u and v of the matrices A and A respectively, which correspond to distinct eigenvalues λ and μ are orthogonal. Thus, the sought subspace (ud+1 , . . . , un ) is orthogonal to the subspace (v 1 , . . . , v d ). Since the dimensions of these subspaces are n − d and d, the subspace (ud+1 , . . . , un ) consists of all vectors that are orthogonal to (v 1 , . . . , v d ). In other words, any non-zero vector that is orthogonal to v 1 , . . . , v d is an eigenvector of the matrix A, which corresponds to the eigenvalue 0. An analogous statement holds for the matrix A . Making use of Lemma 3 of Section 3, we extend the systems of vectors u1 , . . . , ud and v 1 , . . . , v d to systems u1 , . . . , un and v 1 , . . . , v n , so that these systems become biorthonormal and Markov. By the remark just made, these are complete systems of eigenvectors of the matrices A and A , corresponding to the eigenvalues λ1 , . . . , λn . If uk = (u1k , . . . , unk ), v k = (v1k , . . . , vnk ) (k = 1, 2, . . . , n) then the fundamental matrices U = uik n1 and V = vik n1 of A and A are related (by virtue of the biorthogonality of the systems of vectors u1 , . . . , un ; v 1 , . . . , v n ) by the equation V  U = E; hence

V  = U −1 .

We have thus proved Theorem 11. Every sign-definite matrix A = aik n1 of class d+ and rank d has a simple structure. If the eigenvalues of the matrix A are enumerated in order of non-increasing absolute values, then |λ1 | > |λ2 | > · · · > |λd | > λd+1 = · · · = λn = 0, and one can choose a fundamental matrix U = uik n1 such that     i1 . . . ip i1 . . . ip > 0, V >0 U 1 ... p 1 ... p

(48)

(1 ≤ i1 < · · · < ip ≤ n; p = 1, . . . , n),  −1

where V = (U )

. 5. Approximation of a sign-definite matrix by a strictly sign-definite one

In this section we analyze the problem of approximation of a sign-definite matrix by a strictly sign-definite one, and in particular, approximation of a totally non-negative matrix by a totally positive one; we also prove here the existence of strictly sign-definite matrices with prescribed distribution of signs of minors. At the end of this section we give theorems which are in some respect the inverse of the fundamental theorems (Theorem 6 of Section 5, Chap. II and Theorem 5 of Section 4, Chap. IV) which establish the oscillatory properties of the spectrum of an oscillatory matrix and an oscillatory kernel. 1. Theorem 12. An arbitrary sign-definite matrix A = aik  (i = 1, 2, . . . , m; k = 1, 2, . . . , n) of class d and rank r ≥ d can be approximated with any given precision by a strictly sign-definite matrix of class d and rank r.

5. APPROXIMATION OF A SIGN-DEFINITE MATRIX

269

Proof. In (23) of Section 3 of Chap. II we constructed an example of a totally positive matrix Fσ , which depends on the parameter σ and which tends to the unit matrix E as σ → +∞. We denote two such matrices of order m and n respectively (m) (n) by Fσ and Fσ . Using these matrices we define Aσ = Fσ(m) AFσ(n) .

(49)

Then, obviously, lim Aσ = A.

(50)

σ→+∞

On the other hand, we have the identity      i1 i2 . . . ip i1 i2 . . . ip (m) = Fσ Aσ k1 k2 . . . kp α1 α2 . . . αp    β 1 β2 . . . α1 α2 . . . αp Fσ(n) ×A β1 β2 . . . βp k1 k2 . . .

βp kp



(51)

(p = 1, 2, . . . , d). For every p ≤ d, the matrix A has non-zero minors of order p, and all of these minors have the same sign εp ; on the other hand, all minors in A of order > r (m) (n) vanish. If we take into account that Fσ and Fσ are totally positive matrices then it follows from (51) that Aσ has rank ≤ r for every σ > 0, and that all minors of Aσ of any order p ≤ d are different from zero and have sign εp . It is obvious that for sufficiently large σ, the rank of the matrix Aσ is equal to r. This proves the theorem. Corollary. Every totally non-negative matrix of class d and rank r ≥ d can be approximated by a totally positive matrix of class d and rank r. In particular, any totally non-negative matrix of rank r can be approximated by a totally positive matrix of class r and rank r. On the basis of Theorems 9 and 12 we obtain from continuity considerations the following: Theorem 13. If A = aik n1 is a sign-definite matrix of class d and εp is the sign of the minors of p-th order of this matrix (p = 1, 2, . . . , d), then the first d largest eigenvalues (in modulus) of A are real. If the eigenvalues of A are enumerated in the order of non-increasing modulus, then ε2 εp ε1 λ 1 ≥ λ2 ≥ · · · ≥ λp ≥ |λp+1 | ≥ · · · ≥ |λn | ≥ 0. (52) ε1 εp−1 On the basis of Theorems 11 and 12, we obtain from continuity considerations: Theorem 14. A sign-definite matrix A = aik n1 always has real eigenvalues. If the rank of this matrix is r, then by suitable numbering of the eigenvalues, we get λ1 ≥

ε2 εr λ2 ≥ · · · ≥ λr ≥ λr+1 = · · · = λn = 0, ε1 εr−1

(53)

where εp is the sign of the minor of p-th order of the matrix A (p = 1, 2, . . . , r). A special case of this theorem is the following:

270

V. SIGN-DEFINITE MATRICES

Theorem 15. All eigenvalues of a totally non-negative matrix A = aik n1 are real and non-negative. 2. Making use of the corollary of Theorem 12, we can give an interesting generalization of an inequality which follows from the Determinantal Inequality of Section 6, Chap. II. We introduce an abbreviated notation for the minors of (n − 1)-th order of the matrix A = aik n1 :   1 ... i − 1 i + 1 ... n (i, k = 1, 2, . . . , n). Aik = A 1 ... k − 1 k + 1 ... n Using this notation for the minors of (n − 1)-th order, we can write for a totally non-negative matrix the following inequality, which is a special case of the Determinantal Inequality (Section 6, Chap. II):   1 2 ... n ≤ a11 A11 . A 1 2 ... n An essential generalization of this inequality is the following: Theorem 16. If A = aik n1 is a totally non-negative matrix, then   1 2 ... n a11 A11 − a12 A12 + · · · − a1,2p A1,2p ≤ A 1 2 ... n ≤ a11 A11 − a12 A12 + · · · + a1,2q−1 A1,2q−1

 

 n+1 n ; q = 1, 2, . . . , . p = 1, 2, . . . , 2 2

(54)

Proof. It is obvious that this inequality needs to be established only for the case when A has rank ≥ n − 1. In this case, as was shown in Theorem 12, we can approximate the matrix A by a totally non-negative matrix in which all minors of order ≤ n − 1 are positive. Since inequalities (54) are preserved in the limit, it is enough to establish them under the assumption that all the minors of the matrix A of order ≤ n − 1 are positive. Replacing in the matrix A the elements a11 , a12 , . . . , a1s by 0 and denoting the resultant matrix by As (s = 1, 2, . . . , n), we can rewrite (54) as:   1 2 ... n ≥0 (s = 1, 2, . . . , n). (55) (−1)s As 1 2 ... n We shall prove (55) by induction in n. When n = 2, inequality (55) is obvious. Assuming that (55) is valid for matrices of order < n, we write Sylvester’s identity (see Section 2, Chap. I):     3 ... n 1 2 ... n As A 2 ... n − 1 1 2 ... n     2 3 ... n 1 3 ... n (56) = As A 2 3 ... n 1 2 ... n − 1     2 3 ... n 1 3 ... n . −As A 1 2 ... n − 1 2 3 ... n

5. APPROXIMATION OF A SIGN-DEFINITE MATRIX

271

It follows from the induction assumption that    1 3 ... n 1 3 ... s s−1 ≥ 0, and (−1) A2 (−1) As 1 2 ... n − 1 2 3 ...

n n

 ≥ 0.

Therefore, multiplying both sides of (56) by (−1)s and noticing that all minors of the matrix A contained in (56) are positive (since their order is < n), we obtain (55). This proves the theorem. Remark. Theorem 16 is related to one interpolation formula of Academician A.A. Markov [38a,b]. 3. Let A = aik n1 be a strictly sign-definite matrix of class r and rank r. Then (see Theorem 11) A = U λi δik n1 U −1 = U λi δik n1 V  , where λr+1 = · · · = λn = 0, and the matrices U and V satisfy inequalities (48). For arbitrarily given ηr+1 , ηr+2 , . . . , ηn equal to ±1, we define the matrix Aρ (ρ > 0) by Aρ = U μi δik V  ,

(57)

where μ1 = λ1 , . . . , μr = λr ;

μr+1 = ηr+1 ρ, . . . , μn = ηn ρn−r .

Evidently, lim Aρ = A.

ρ→0

On the other hand, it follows from (57) that   i1 . . . ir+s Aρ = k1 . . . kr+s     k1 i1 . . . ir+s V μα1 μα2 . . . μαr+s U = α1 . . . αr+s α1 α1 r. In particular, a totally non-negative matrix can be always approximated by a totally positive matrix. The arguments used to prove Theorem 17, make it also possible to prove the following theorem: Theorem 18. Let there be given two arbitrary biorthonormal Markov systems of vectors uj = (u1j , . . . , unj ) (j = 1, 2, . . . , n) and v j = (v1j , . . . , vnj ) (j = 1, 2, . . . , n). Then, for every given sequence of signs ε1 , ε2 , . . . , εr (1 ≤ r ≤ n) there always exists a matrix A = aik n1 of rank r, having the following properties: 1◦ every minor or order p ≤ r is different from zero and has sign εp ; 2◦ the matrices U = uik n1 and V = vik n1 are fundamental matrices for A and A respectively. Proof. It is easy to see that for sufficiently small ρ > 0, the matrix   0 ... 0 0 ... 0  η1 ρ   2 0 0 ... 0  0 η2 ρ . . .     ..............................   2 0 . . . ηr ρ 0 . . . 0  V  , Aρ = U  0   0 ... 0 0 ... 0  0     ..............................   0 0 ... 0 0 ... 0

(59)

where ηk = εk εk−1 (k = 1, 2, . . . , r; ε0 = 1) has properties 1◦ and 2◦ , indicated in the theorem. 4. By means of analogous arguments we can also prove the following theorem. Theorem 19. If a matrix A = aik n1 has n non-zero real eigenvalues λ1 , λ 2 , . . . , λ n whose absolute values are distinct: |λ1 | > |λ2 | > · · · > |λn | > 0,

(60)

and the eigenvectors uj = (u1j , u2j , . . . , unj )

and

v j = (v1j , v2j , . . . , vnj )

(j = 1, 2, . . . , n)

of the matrices A and A , corresponding to these eigenvalues, form two Markov systems, then some power of the matrix A is totally positive. Proof. In analogy with the proof of Theorem 10, we can normalize the vectors u1 , u2 , . . . , un and v 1 , v 2 , . . . , v n so that they become biorthonormal: UV  = E

(U = uik n1 , V = vik n1 ),

and satisfy simultaneously the inequalities    i1 i2 . . . i1 i2 . . . ip > 0, V U 1 2 ... p 1 2 ...   1 ≤ i1 < i2 < · · · < ip ≤ n . p = 1, 2, . . . , n

ip p

 >0 (61)

5. APPROXIMATION OF A SIGN-DEFINITE MATRIX

273

Furthermore,

(V  = U −1 ). A = U λi δik V  Let us put now B = A , where m is a positive integer. Then 2m

 B = U λ2m i δik V ,



and B

i1 k1 =

i2 k2

... ... 

ip kp



2m 2m λ2m α1 λα2 . . . λαp

1≤α1 0:  0 for i = k k k  k k i k Au = λk u , A v = λk v , (u v ) = δik = 1 for i = k (3) (i, k = 1, 2, . . . , n). The elements of the matrix A = aik n1 satisfy the relations n 

aik ≥ 0,

aij = 0

(i, k = 1, 2, . . . , n).

(4)

j=1

Furthermore, we can assume ui1 > 0, vi1 > 0

(i = 1, 2, . . . , n).

For an arbitrary vector x we have the expansion x=

n 

(xv k )uk .

(5)

k=1

Applying a linear transformation with matrix Am to both sides of this equation, we obtain m  m k k A x= λm (m = 1, 2, . . . ). k (xv )u k=1

It follows that as m → ∞



n λ−m 1 A x

1

1

λ2 λ1

m

= (xv )u + (xv 2 )u2 + . . .  m λn (xv n )un → (xv 1 )u1 . + λ1

Let now x be an arbitrary vector with positive coordinates. Then c = (xv 1 ) > 0 and, by virtue of (4), all coordinates of all vectors (m)

(m)

x(m) = Am x = (x1 , x2 , . . . , xn(m) )

(m = 1, 2, . . . )

are positive: (m)

xi

>0

(i = 1, 2, . . . , n; m = 1, 2, . . . ).

(6)

METHOD OF APPROXIMATE CALCULATION OF EIGENVALUES

277

In coordinate form, relation (6) can be written as λ−m 1 xi

(m)

→ cui1

as m → ∞;

hence (m)

xi

(m−1)

xi (m)

→ λ1

as

m → ∞,

(7)

(m)

x1 :x2 : . . . : xn(m) → u11 : u21 : . . . : un1 .

(8)

According to the lemma, for every m (m)

min

1≤i≤n

(m)

xi

(m−1)

xi

xi

≤ λ1 ≤ max

(m−1)

1≤i≤n

.

(9)

xi

Thus, the largest eigenvalue λ1 and the coordinates of the corresponding eigenvector u1 = (u11 , u21 , . . . , un1 ) of an oscillatory matrix A can be computed by the following rule: Let x = (x1 , x2 , . . . , xn ) be an arbitrary vector with positive coordinates and ' x(1) = Ax, x(2) = Ax(1) , . . . , x(m) = Ax(m−1) , . . . (10) (m) (m) x(m) = (x1 , x2 , . . . , xn(m) ) (m = 1, 2, . . . ). Then

(m)

1)

xi

λ1 = lim

m→∞

(i = 1, 2, . . . , n),

(m−1)

(11)

xi

and for every m, (m)

lm = min

1≤i≤n

xi

(m−1)

xi

(m)

≤ λ1 ≤ max

xi

= Lm ;

(12)

(i = 1, 2, . . . , n).

(13)

1≤i≤n

(m−1)

xi

(m)

2)

xi ui1 = lim m→∞ x(m) u11 1

Inequalities (12) make it possible to estimate λ1 from above and from below. Remark. It can be shown that this rule remains in force under considerably more general conditions, namely: it holds for an arbitrary matrix A with nonnegative elements, a sufficiently high power of which has positive elements. If such a matrix has simple structure (for example, if the matrix is symmetric), then the proof already given carries through. In the general case, it is necessary to use the representation3 of Ax as Ax = λ(v, x)u + A1 x,

(14)

where λ > 0 is the largest eigenvalue (in modulus) and u, v, are the corresponding eigenvectors: Au = λ1 u, A v = λ1 v (u, v) = 0, while A1 is a linear transformation with the property A1 u = A1 v = 0. 3 The

possibility of such a representation can be proved using Perron’s theorem.

(15)

278

SUPPLEMENT I

The geometric interpretation of (14) is that with respect to the transformation x = Ax, the space En of n-dimensional vectors is a direct sum of two invariant subspaces: the one-dimensional one {λu} and the hyperplane (v, x) = 0. Therefore the eigenvalues of A1 are the eigenvalues A different from λ and, in addition, the number 0. It follows from (14) and (15) that Am x = λm (v, x)u + Am 1 x

(m = 1, 2, . . . ),

and since all the eigenvalues A1 have moduli less than λ, lim λ−m Am 1 x = 0,

m→∞

from which the statement follows. 3. Let us calculate the second eigenvalue λ2 and the corresponding eigenvector u2 = (u12 , u22 , . . . , un2 ) of an oscillatory matrix A. Consider the associated matrix A2 . Its elements will be the minors        i 0, B0

(6)

and the polynomials A(1) (λ) and B (1) (λ) (suitably normalized) satisfy the relations: A(1) (λ) = A(λ) − aB(λ),

(7 )

B (1) (λ) + bλA(1) (λ) = B(λ).

(7 )

Let (1)

(1)

(1)

(1)

(1)

A(1) (λ) = A0 λn−1 + A1 λn−2 + · · · + An−1 ; (1)

B (1) (λ) = B0 λn−1 + B1 λn−2 + · · · + Bn−1 . Comparison of the coefficients of λn in (7 ) gives (1)

bA0 = B0 > 0. On the other hand, comparison of the coefficients of λn−1 in (7 ) yields (1) A0

= A1 − aB1 = A0

n 

λi − aB0

1

n 

μi = A 0

n 

1

(λi − μi ) > 0.

1

Thus, for a given positive pair {A(λ), B(λ)} we can uniquely define polynomials of (n − 1)-th degree A(1) (λ) and B (1) (λ) and the numbers a and b from relations (7 ) and (7 ); the numbers a and b will be positive. To prove that the polynomials A(1) (λ) and B (1) (λ) form a positive pair, we rewrite relation (7 ) in the form A(1) (λ) A(λ) = −a B(λ) B(λ)

(= F (λ))

and investigate the behavior of the right hand side F (λ) as λ varies from −∞ to 0. By (1), (2) and (3), we have lim

λ→−μj +0

A(λ) = +∞, B(λ)

lim

λ→−μj −0

A(λ) = −∞, B(λ)

and consequently F (−μj + 0) = +∞,

F (−μj − 0) = −∞ (j = 1, 2, . . . , n).

Thus, within each interval (−μj+1 , −μj )

(j = 1, 2, . . . , n − 1)

3 The degree of the denominator in the second summand of the right hand side of the preceding formula is n (see the following formula (7 )), and the degree of the numerator does not exceed n − 1. Consequently, as λ → ∞, this summand tends to zero.

286

SUPPLEMENT II

the function F (λ) changes sign, and this implies that the polynomial A(1) (λ) has a root λ = −λj inside this interval (j = 1, 2, . . . , n − 1); i.e., n−1 

(λ + λj ),

(8)

(j = 1, 2, . . . , n − 1).

(9)

(1)

A(1) (λ) = A0

j=1

with

μj < λj < μj+1 We now rewrite (7 ) in the form

B (1) (λ) B(λ) = −bλ + (1) (1) A (λ) A (λ)

(= F1 (λ))

and study the behavior of F1 (λ), in a way similar to that of F (λ). From (8) and (9) we easily find that B(λ) = −∞, λ→−λj +0 A(1) (λ) lim

B(λ) = +∞, λ→−λj −0 A(1) (λ) lim

so that F1 (−λj + 0) = −∞,

F1 (−λj − 0) = +∞ (j = 1, 2, . . . , n − 1).

Consequently, in each interval (−λj+1 , −λj )

(j = 1, 2, . . . , n − 2)

the numerator B (1) (λ) has at least one zero. Noticing that F1 (−λ1 + 0) = −∞, and B (1) (0) B(0) = F1 (0) = (1) > 0, A(1) (0) A (0)

(10)

we also conclude that B (1) (λ) has a zero in the interval (−λ1 , 0). Thus n−1  (1) (1) B (λ) = B0 (λ + μj ), j=1

where

0 < μ1 < λ1 < μ2 < λ2 < · · · < μn−1 < λn−1 . It also follows from (10) that (1)

B0

> 0.

This proves the lemma. 3. We notice that the polynomials A(1) (λ) and B (1) (λ) are obtained from A(λ) and B(λ) by rational operations – by applying twice the first step, consisting in ordinary division of polynomials according to the following scheme: A(λ) A(1) (λ) 1 1 = an + = an + = an + . B(λ) B(λ) B(λ) B (1) (λ) bn λ + (1) A(1) (λ) A (λ) Here and in what follows we write an and bn instead of a and b.

REMARKABLE PROBLEM FOR A STRING WITH BEADS

287

Representing similarly the ratio A(1) (λ)/B (1) (λ): A(1) (λ) = an−1 + B (1) (λ)

1 B (2) (λ) bn−1 λ + (2) A (λ)

,

and substituting in (4), we find that A(λ) = an + B(λ)

1

.

1

bn λ +

1

an−1 +

bn−1 λ +

B (2) (λ) A(2) (λ)

Continuing this way, after n steps we expand the ratio A(λ)/B(λ) into a continued fraction, in which the last incomplete quotient will be the fraction A(n) (λ)/B (n) (λ), the numerator and denominator of which make a positive pair of zero degree. Consequently, A(n) (λ) ≡ a0 > 0. B (n) (λ) We have thus proved the following theorem of Stieltjes: Theorem. For every positive pair {A(λ), B(λ)} of degree n ≥ 1 there exists one and only one expansion of the ratio A(λ)/B(λ) into a continued fraction of the form A(λ) 1 = an + . (11) 1 B(λ) bn λ + 1 an−1 + 1 bn−1 λ + · · · + 1 a1 + 1 b1 λ + a0 In this expansion we always have ai > 0

(i = 0, 1, . . . , n);

bi > 0

(i = 1, 2, . . . , n).

(12)

Stieltjes has also shown the converse, that every continued fraction (11) with arbitrary positive numbers ai and bi is equal to the ratio A(λ)/B(λ), where A(λ) and B(λ) is a positive pair.4 We will obtain this result as a byproduct of our solution of problem I. 4. Returning to problem I, we recall (see Section 2, Chap. III) that the kinetic energy T and the potential energy V of a stretched string S bearing n beads with masses m1 , m2 , . . . , mn have the following expressions: 1 mi y˙i2 , 2 i=1 n

T =

V =

n σ1 (yi+1 − yi )2 , 2 i=0 li

4 A continued fraction is reduced to the form A(λ)/B(λ) by formally performing all the operations indicated in the fraction, without additional multiplication or cancellation of numerators and denominators by a common factor. Thus the numerator A(λ) and the denominator B(λ) are polynomials with positive coefficients of the 2n + 1 variables ai (i = 0, 1, . . . , n) and bi λ (i = 1, 2, . . . , n).

288

SUPPLEMENT II

where, assuming stationary ends, we have y0 = yn+1 = 0. We note that by changing the unit of length we can always make the tension σ equal to one, in which case l and all li (i = 0, 1, . . . , n) change their numerical values by factor of σ. Therefore, without loss of generality, to simplify our formulas we shall for the most part assume that σ = 1. In any formula obtained for σ = 1, replacing l by l/σ and li by li /σ (i = 1, 2, . . . , n) will make this formula suitable for the general case. Writing down the Lagrange equations for the string S and then putting in these equations yi = ui sin(pt + α) (i = 1, 2, . . . , n) we obtain for the frequency p and for the displacement amplitudes ui (i = 1, 2, . . . , n) the following equations (compare with equations (14) in Sect. 2, Chap. III): ui − ui+1 ui − ui−1 + − mi p2 ui = 0 (i = 1, 2, . . . , n), li li−1 u0 = 0, un+1 = 0.

(13) (14)

Since u0 = 0, we can successively determine the values of ui (i = 2, 3, . . . , n + 1) from the recurrence relations (13): ui = R2i−2 (λ)u1

(i = 1, 2, . . . , n + 1; λ = −p2 ),

(15)

where R2i−2 (λ) is a polynomial of degree i − 1 (i = 1, 2, . . . , n + 1). The frequencies p1 < p2 < · · · < pn of the string are then determined by the equation R2n (λ) = 0 (λ = −p2 ). Let us introduce, furthermore, the polynomials R2i−1 (λ) =

R2i (λ) − R2i−2 (λ) li

(i = 1, 2, . . . , n),

(16)

so that

ui+1 − ui = R2i−1 (λ)u1 (i = 1, 2, . . . , n). li Thus, for fixed values of λ, those polynomials with even indices R0 , R2 (λ), R4 (λ), . . . , R2n−2 (λ)

give, up to a constant factor, the displacement amplitudes of the beads m1 , m2 , . . . , mn of the corresponding harmonic oscillation; polynomials with odd indices R1 (λ), R3 (λ), . . . , R2n−1 (λ) give, up to a constant factor, the angles of inclinations (more accurately, the slopes) of the successive segments li (i = 0, 1, . . . , n) of the string.

REMARKABLE PROBLEM FOR A STRING WITH BEADS

289

The polynomials Rk (k = −1, 0, 1, 2, . . . , 2n) can be determined successively from the recurrence formulas R2i−1 (λ) = λmi R2i−2 (λ) + R2i−3 (λ), R2i (λ) = li R2i−1 (λ) + R2i−2 (λ) 1 (i = 1, 2, . . . , n; R−1 (λ) = , R0 (λ) ≡ 1). l0

(17)

The first system of relations is obtained by substituting expressions (15) for the coordinates ui (i = 2, 3, . . . , n) into (13), while the second system is equivalent to (16). Consequently, R2i (λ) 1 1 = li + = li + . 1 R2i−1 (λ) R2i−1 (λ) mi λ + R2i−2(λ) R2i−2 (λ) R2i−3(λ) From this we conclude that R2n (λ) = ln + R2n−1 (λ)

1

.

1

mn λ +

(18)

1

ln−1 +

1

mn−1 λ + . . .

1

l1 +

m1 λ +

1 l0

5. Let us consider the case that the left end of the string is fastened and the right end is allowed to slide freely in a direction perpendicular to the equilibrium position of the string. Equations (13) still remain in force, but in place of the “boundary” condition un+1 = 0 we have the condition un+1 = un . According to (15), with λ = −p2 , this condition implies R2n (λ) = R2n−2 (λ) or, which is the same, R2n−1 (λ) = 0. p1

p2

(19)

pn

< ··· < be the frequencies of the string now under consideration, Let < with the sliding end; i.e., let λj = −pj

2

(j = 1, 2, . . . , n)

be all roots of the equation (19). When the right end is fixed, the potential energy V of the string is given by V =

n 1  1 (yi+1 − yi )2 2 i=0 li

(y0 = yn+1 = 0),

290

SUPPLEMENT II

and when the right end is allowed to slide, it is given by V =

n 1  1 (yi+1 − yi )2 (y0 = 0, yn = yn+1 ). 2 i=0 li

Thus the form V is obtained from the form V  by adding one square (in the first case the fastening is more rigid than in the second5 ). According to Theorem 15 of Chap. I (which gives here a more accurate result than Law II of Section 1 of Chap. III), p1 ≤ p1 ≤ p2 ≤ · · · ≤ pn ≤ pn . It is easy to see that in fact all these inequalities are strict. Indeed, the contrary assumption implies that the polynomials R2n (λ) and R2n−1 (λ) have a common root λ0 . But then, by the recurrence relations (17), λ0 would be a common root of R2n−2 (λ), R2n−3 (λ), . . . , R0 (λ) which is impossible since R0 (λ) = 1. Thus, p1 < p1 < p2 < · · · < pn < pn . (20) By the recurrence relations (17), all the coefficients of the polynomial Ri are positive. Thus ⎫ n  ⎪ 2 ⎪ R2n = C (λ + pj ) (C > 0), ⎪ ⎪ ⎪ ⎬ j=1 (21) n ⎪  ⎪  2  ⎪ (λ + pj ) (C > 0).⎪ R2n−1 = C ⎪ ⎭ j=1

Taking (20) into account, we conclude that the polynomials R2n and R2n−1 form a positive pair. Since the masses m1 , . . . , mn and the lengths l0 , l1 , . . . , ln can be chosen arbitrarily, this proves the converse theorem of Stieltjes mentioned in Subsection 3. 6. Now we have everything that is needed to solve our problem. Starting out with given values p1 < p1 < p2 < p2 < · · · < pn < pn , we set up the polynomials A(λ) = A0 B(λ) = B0

n 

(λ +

i=1 n 

(λ +

p2i ),

(22)

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪

2 ⎪ ⎪ pi ),⎪ ⎭

(23)

i=1

choosing A0 > 0 and B0 > 0 arbitrarily. By virtue of (22), these polynomials form a positive pair and consequently expansion (11) holds. If we could find a string with beads which solves the problem then, according to (21), the ratio of polynomials R2n (λ) and R2n−1 (λ) for this string would be R2n (λ) A(λ) =ρ , R2n−1 (λ) B(λ) 5 See

the end of Section 3 of Chap. III.

REMARKABLE PROBLEM FOR A STRING WITH BEADS

291

where ρ is a positive factor. But it is easily seen that, when the left hand side of (11) is multiplied by ρ, all numbers ai are multiplied by ρ, and all numbers b1 are divided by ρ. Taking this into account and recalling (18), we conclude that the sought mi and li are li = ρai

(i = 0, 1, . . . , n),

mi = ρ−1 bi

(i = 1, 2, . . . , n).

We have not yet used the conditions l0 + l1 + · · · + ln = l, where l is the given length of the string. This condition gives ρ=

l . a0 + a1 + · · · + an

If we recall the comments at the beginning of Subsection 1 concerning the tension σ, then for σ = 1 the final formulas for li and mi will be: ⎫ ai l (i = 0, 1, . . . , n),⎪ li = ⎬ a0 + a1 + · · · + an (24) σ ⎭ (i = 1, 2 . . . , n).⎪ mi = (a0 + a1 + · · · + an )bi l This solves the problem. We note now that by an appropriate choice of the coefficients A0 and B0 in (23), we can simplify the formulas (24). Indeed, if the quotient A(λ)/B(λ) of the two polynomials A(λ) = A0 λn + · · · + An ,

B(λ) = B0 λn + · · · + Bn

(n ≥ 1)

admits an expansion (11), then by (11) with λ = 0 we have An = a0 + a1 + · · · + an . Bn Therefore, if we put  n   λ 1+ 2 , A(λ) = pi i=1

 n   λ B(λ) = 1 + 2 , pi i=1

(25)

we shall have An = Bn = 1,

a0 + a1 + · · · + an = 1.

Thus, we obtain the following rule for solving problem I: Use (25) to define a positive pair {A(λ), B(λ)} and expand it as in (11). Then 

l i = ai l mi = bi σ/l

(i = 0, 1, . . . , n), (i = 1, 2, . . . , n).

7. We can solve the following problem is a similar way.

(26)

292

SUPPLEMENT II

Problem II. Consider a string with a sliding left end, bearing n beads, one of which is placed on the left end 6 . The length l and the tension σ of the string are specified. In addition, let 2n numbers be specified, 0 = p1 < p1 < p2 < p2 < p3 < · · · < pn < pn . It is required to find the masses and the arrangement of the beads such that when the right end is fixed the frequencies of the string are p1 , p2 , . . . , pn , and when the right end is allowed to slide freely the frequencies are p1 , p2 , . . . , pn . In this case, the equations for the coordinates of an amplitude vector and for a frequency are written in the form (σ = 1): ⎫ u1 − u2 ⎪ ⎪ − m1 p2 u1 = 0, ⎬ l1 (27) ui − ui+1 ui − ui−1 ⎪ + − m i p 2 ui = 0 (i = 2, 3, . . . , n);⎪ ⎭ li li−1 to these equations one must add one of the two “boundary” conditions on the right end: un+1 = 0 or un = un+1 , depending on how it is fastened. Equations (27) are obtained by substituting the expression yi = ui sin(pt + α) (i = 1, 2, . . . , n) to the Lagrange equations for the system. They may, however, also be obtained from (13), to which one adds the condition u0 = u1 instead of the boundary condition u0 = 0 on the left end. This is explained by the fact that the constraint imposed on the small ring m1 can also be produced by removing the wire passing through it and joining it with a stretched string of arbitrary length l0 to a second, massless ring, which is allowed to slide freely on a parallel wire. From the recurrence relations (13) we again obtain ui = Q2i−2 (λ)u1

(i = 1, 2, . . . , n + 1; λ = −p2 ; Q0 ≡ 1),

where Q2i (λ) is a polynomial of degree i (i = 0, 1, . . . , n). In analogy with the preceding, we introduce the polynomials Q2i−1 (λ) =

Q2i (λ) − Q2i−2 (λ) li

(i = 1, 2, . . . , n),

and obtain the recurrence relations: Q2i−1 (λ) = λmi Q2i−2 (λ) + Q2i−3 (λ), Q2i (λ) = li Q2i−1 (λ) + Q2i−2 (λ) (i = 1, 2, . . . , n; Q0 (λ) ≡ 1, Q−1 (λ) ≡ 0). These relations coincide with (17) for polynomials Ri , but whereas in that case R−1 (λ) = 1/l0 , in the present case we have Q−1 (λ) ≡ 0. 6 This bead is best visualized as a small ring of mass m , sliding freely on an ideally smooth 1 wire passing through it (see Section 2, Chap. III).

REMARKABLE PROBLEM FOR A STRING WITH BEADS

293

It is easy to see that Q2n (λ) = ln + Q2n−1 (λ)

1 1

mn λ +

1

ln−1 +

1

mn−1 λ + · · · + l1 +

1 m1 λ

From the equation (λ = −p2 )

Q2n (λ) = 0

we obtain the frequencies p1 < p2 < · · · < pn of the string under consideration when its right end is fixed, and from the equation (λ = −p2 )

Q2n−1 (λ) = 0

we obtain the frequencies 0 = p1 < p2 < · · · < pn for a sliding right end. As in the preceding case, we establish that p1 < p1 < p2 < p2 < · · · < pn < pn . The solution of problem II consists of the following: Introduce the polynomials C(λ) = C0

n 

(λ + p2j ),

D(λ) = D0

j=1

n 

(λ + pj ) 2

(C0 , D0 > 0).

j=1

Then find the expansion C(λ) = an + D(λ)

1

.

1

bn λ + an−1 +

1 bn−1 λ + · · · +

1 a1 +

Then

(28)

1 b1 λ

⎧ ⎨ li =

ai l a1 + a2 + · · · + an (l = 1, 2, . . . , n). ⎩ m = σ (a + a + · · · + a ) b i 1 2 n i l The possibility of the expansion (28) and the fact that all the ai and bi (i = 1, 2, . . . , n) are positive in this expansion, is proved analogously to what was done for the expansion (11). Remark. If the polynomials C(λ) = C0 λn + · · · + Cn ,

D(λ) = D0 λn + · · · + Dn−1 λ

294

SUPPLEMENT II

satisfy (28), then D(λ) = λC(λ)

1

.

1

an λ +

1

bn +

1

an−1 λ + · · · +

a1 λ +

1 b1

Letting λ tend to zero, we find that Dn−1 = b1 + b2 + · · · + bn . Cn Therefore, if we specify instead of l the sum of all the masses M = m 1 + m2 + · · · + mn , then, putting C(λ) =

n   j=1

 λ 1+ 2 , pj

 n   λ D(λ) = λ 1 + 2 pj j=2

and using the expansion (28), we obtain the quantities li and mi (i = 1, 2, . . . , n) from the formulas  mi = M bi (i = 1, 2, . . . , n) li = σai /M 8. Mechanical interpretation of the Stieltjes’ investigations. Let us examine the continued fraction 1

.

1

m1 λ +

(29)

1

l1 + m2 λ +

1 l2 +

1 .. .

The numerators and denominators of the successive convergents of this continued fraction will satisfy the same recurrence relations as the polynomials Ri (λ) and Qi (λ) (i = 1, 2, . . . ) considered above. Furthermore, since the first two polynomials Q1 (λ) = m1 λ,

and Q2 (λ) = l1 m1 λ + 1

correspond, respectively, to the denominators of the first and second convergents for (29), we see that in general the polynomials Qi (λ)(i = 1, 2, . . . , n) are nothing but the denominators of the successive convergents to the continued fraction (29). Let Pi /Qi be the i-th convergent for (29): P1 1 , = Q1 m1 λ

P2 l1 ,... = Q2 l 1 m1 λ + 1

REMARKABLE PROBLEM FOR A STRING WITH BEADS

295

It is easy to show that the polynomials Ri (λ) are expressed in terms of polynomials Pi and Qi by the formula Ri (λ) =

1 Pi (λ) + Qi (λ) l0

(i = 1, 2, . . . ).

Stieltjes’ memoir [52] which gained him so much fame, is devoted to the investigation of infinite continued fractions of the form (29), where mi and li (i = 1, 2, 3, . . . ) are arbitrary positive numbers. Apparently Stieltjes was not aware of the above mechanical interpretation that can be given to his investigations. Once this interpretation is made, many subtle algebraic and function-theoretic theorems of Stieltjes acquire a straightforward mechanical evidence. Thus, for example, on pp. 15-20 of his memoir, Stieltjes proves that the roots λ1 , λ2 , . . . , λn of the denominator Qm (λ) (m = 2n or 2n−1) of the m-th convergent for (29) are non-decreasing functions of the mi and li (i = 1, 2, . . . , n). If we recall that the roots λi (i = 1, 2, . . . , n), taken with the opposite sign, are the squares of the frequencies of a string fastened in a definite manner at the ends and bearing n beads of masses m1 , . . . , mn , which separate the string into n successive segments of lengths l1 , l2 , . . . , ln , then Stieltjes’ statement becomes evident. Indeed, if one increases at least one of the masses mi , the kinetic energy of the string increases, and if one increases at least one of the lengths li , the potential energy of the string decreases, as is obvious from the expressions for these two energies. It remains to recall laws II and III of Section 1 of Chap. III. Stieltjes shows that it may happen that one of the roots λj does not depend on one of the masses mi or on one of the lengths lk . This is also clear from the point of view of mechanics. If in a harmonic oscillation of a string, corresponding to a root λj , one of the nodes accidentally happens to be at mi , then no matter how this mass is changed, the number pj = −λj will always remain a frequency of the string. In exactly the same manner, if in a harmonic oscillation some segment of the string lk oscillates parallel to itself, then this same oscillation will be possible with any change in the length lk ; i.e., λj is independent of lk . Let us sketch mechanical interpretation of certain results of Stieltjes’ analysis of the properties of an infinite continued fraction (29). Stieltjes’ naturally singles out the case when the fraction (29) diverges for at least one (and thus for all) positive λ. This case is characterized by the condition ∞  i=1

mi < ∞,

∞ 

li < ∞.

i=1

(30)

 We identify this case with a string S of finite length l = li , on which an infinite sequence of beads of masses m , m , . . . , m , . . . with total finite mass M = 1 2 n  mi is placed, where the positions of the beads converge to the right endpoint. Furthermore, the first bead is placed at the left end and can slide freely in a direction perpendicular to the equilibrium position of the string. Stieltjes shows that in case (30), as n → ∞, the polynomials (2n)

Q2n (−λ) = 1 − c1

λ + ...

296

SUPPLEMENT II

converge uniformly in each bounded part of the plane to an entire function D(λ), while the polynomials (2n−1)

Q2n−1 (−λ) = −c1

λ + ...

(2n−1)

(c1

=

n 

li )

1

converge uniformly in each bounded part of the plane to another entire function −lΔ(λ) = −lλ + . . . It is easy to explain the meaning of these entire functions D(λ) and Δ(λ). Let us imagine that the right end of the string S is fixed. Then the squares λ of its frequencies are the eigenvalues of the integral equation (see Chap. IV):  l ϕ(x) = λ K(x, s)ϕ(s) dσ(s), (31) 0

where K(x, s) is the influence function of the string S with fixed right end and sliding left end; i.e., (see (72), Chapter III, with T = 1)  l − s (x ≤ s) K(x, s) = (32) l − x (x ≥ s), and σ(x) (σ(0) = 0) is the distribution function of the masses on the string S; i.e., the step function with jumps exactly at xn =

n−1 

li

(n = 1, 2, . . . ; x1 = 0),

i=1

with σ(xn + 0) − σ(xn − 0) = mn (n = 1, 2, . . . ). It is easy to guess that the entire function D(λ) is nothing but the Fredholm determinant of the integral equation (31). Once this is established, one can give an explicit expression for all coefficients of the entire function D(λ) which was not known to Stieltjes. In view of the special structure (single-parity) of the kernel K(x, s), the usual Fredholm expressions for D(λ) can be replaced by simpler ones, as we have already seen in Section 9 of Chapter IV. Therefore,  ∞  λ D(λ) = 1− 2 , pj j=1 where pj (j = 1, 2, . . . ) are the frequencies of the string S with fixed right end. Similarly  ∞  λ 1− 2 , Δ(λ) = λ p j j=2 where pj (j = 1, 2, . . . ; p1 = 0) are the frequencies of the string S with freely sliding right end. One can show that Δ(λ)/λ is the Fredholm determinant of an integral equation (31) in which the kernel K(x, s) is defined not by by (32) but by some other formulas; this kernel is a generalized influence function (generalized Green function) of the string S with two sliding ends, but we will not dwell on this.

REMARKABLE PROBLEM FOR A STRING WITH BEADS

297

We denote by {ϕj (x)}∞ 0 an orthonormal system of fundamental functions of equation (31) (in other words, the amplitude functions of the string S with fixed right end):  l ϕj (x)ϕk (x)dσ(x) = δjk (j, k = 0, 1, . . . ). 0

Then the resolvent Γ(x, s; λ) =

∞  ϕj (x)ϕj (s) λ − p2j j=0

(λ = p2 ; x, s ∈ Iσ )

will give the amplitude deflection of the forced oscillations of the string S under the action of the unit force which has frequency p and is concentrated at the point s. It turns out that as n → ∞ the polynomials P2n (−λ) also converge uniformly in each bounded part of the plane to a function E(λ), and ∞ E(λ)  ϕ2j (0) = = Γ(0, 0; λ). D(λ) j=0 λ − p2j

Thus, E(λ) = D(0, 0; λ), where D(x, s; λ) is the Fredholm minor of equation (31). For a string S with two moving ends, the notion of resolvent is also defined, and in this case we have ∞  ψj2 (0) E(λ) = , Δ(λ) j=1 λ − pj 2 where E(λ) is the uniform limit in each bounded part of the plane of polynomials P2n−1 (λ), and {ψj (x)} an orthonormal system of amplitude functions of the string S with free ends. The mechanical interpretation of Stieltjes’ research becomes more complicated if one or both of the conditions (30) is not satisfied. In this case either the string has an infinite length, or the total mass of the beads is infinite, or both. One can, however, develop a theory of oscillations of a semi-infinite (or even infinite) string, each finite part of which is loaded by a finite mass which may be distributed both continuously and discretely. In this theory, all the results of Stieltjes’ memoir [52] will be obtained as special cases.

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Remarks Chapter I Chapter I contains results from the theory of matrices and quadratic forms (see, for example, the texts by P.A. Shirokov [51], R. Courant and D. Hilbert [8], A.I. Mal’tsev [37], I.M. Gel’fand [17], D.A. Grave [19], and others). The proof by induction of the generalized Hadamard inequality given in Section 8 is close to the proof of the more general theorems discussed in Section 6 of Chap. II. Let us point out a recent generalization of the Hadamard theorem in a different direction, obtained by M.K. Fage [12].

Chapter II 1. The theory of oscillatory matrices was first developed by the authors [16a,b,c]. In this edition of the book, the development of this theory in Chapter II differs little from that of in [16c]. Let us list some of the supplements to this paper which are contained in Chapter II. a) As an introduction to the general theory of oscillatory matrices, we give in Section 1 an independent development of the basic properties of normal Jacobi matrices – the forerunners of the oscillatory matrices. The basic facts of the theory of Jacobi matrices were already obtained by Sturm, but he did not publish his results (see [7b] on this topic, as well as the footnote at the beginning of Section 2 of Chapter III). Apparently the first to revive Sturm’s results was E. Routh [45a,b]. In writing Section 1, the authors also used papers by M.G. Krein [30b,c]. b) The Determinantal Inequality (76) for totally non-negative matrices was first obtained by the authors in [16a]. This paper also contains the application of this inequality to the theory of integral equations with totally non-negative kernels, to which we devote a separate remark in [16a]. In the first edition of this book we indicated some generalizations of this inequality, which contain the generalized Hadamard inequality. A recent paper by D.M. Kotelyanskii [29], containing further investigations on this problem, has led the authors to reconsider their original method with the goal of obtaining conclusions of maximal generality. As a result we obtained the general theorems of Section 6. c) In Section 10 we added the propositions from the paper by M.G. Krein [30b] on the behavior of the spectrum of a normal Jacobi matrix with varying coefficients. Theorems 16 and 17 of Section 10 generalize these propositions. 2. The paper by D.M. Kotelyanskii [29] presents an interesting investigation about the structure of arbitrary totally non-negative matrix; in particular it is 299

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shown in this paper that every non-singular symmetric totally non-negative matrix is a direct sum of several oscillatory matrices. 3. After Perron’s work [40], investigations on matrices with non-negative elements were undertaken by Frobenius [14a,b]. Recently interest in these matrices has become quite lively in connection with a study of Markov chains in the theory of probability and Soviet mathematicians have obtained many substantial results about these matrices. We mention the book by V.I. Romanovskii [44] on this subject where further literature is given, and also the recent articles by N.A. Dmitriev and E.B. Dynkin [11a,b], V.G. Kalagastov [26], Kh. R. Suleimanova [54], and F.I. Karpelevich [27]. Chapter III 1. The main ideas of this chapter belong to M.G. Krein. As indicated in the introduction, he observed for the first time in 1934 [30e,f] (from considerations other than those given here) that the influence function of a segmental continuum, under the most frequently used conditions of fastening of the ends, is a Kellogg kernel (the theory of which Krein derived without knowing the works of Kellogg). It was precisely this circumstance which served as an impetus to further investigations by the authors on the theory of oscillatory matrices [16a,b,c], kernels [30g,h], and differential operators [30d,i,k,l] (see also P.D. Kalafati [25,b,c]). Later, in 1937, M.G. Krein established the mechanical interpretation of the oscillatory character of the influence function, which was formulated in Theorem 2 of Section 6. In this connection Krein noted the significance of Rolle’s theorem in these considerations. In the first edition of this book, Theorem 2 of Section 6 was proved by purely algebraic means, now treated in Chapter V. In this edition we revive the original proof by M.G. Krein. The simple proof (based on Theorem 2 and Rolle’s theorem) that, under all usual conditions of end fastening of an elastic rod, the influence function of the rod is an oscillatory kernel (Theorem 4, Section 8) is given in this book for the first time. In the first papers by M.G. Krein, this was proved only for the most frequently used conditions of fastening of the ends of a rod. The methods were based on certain specific expressions for the influence function of the rod. The latter method of proof is interesting because it permits an investigation of the antinodes and the inflection points of the natural oscillations of rods (see [30g]) under various boundary conditions. 2. As was shown in the case of an elastic continuum loaded with n masses (Chapter III, Section 7), as well as for the case of an arbitrary loaded continuum (Chapter IV, Section 4), the oscillatory character of the influence function of a continuum implies all the oscillatory properties of its natural oscillations. We mention that theorems on the simplicity of the spectrum of frequencies of a rod and on the number of nodes of its amplitude functions were first proved by O. Davidoglou [9a,b] (by a method of E. Picard). Davidoglou considered only the simplest cases of fastening of the ends and assumed that the mass has a continuous density. Theorems of this kind, under the foregoing assumption, were obtained by another method by S.A. Janczewski [22] for a very large number of boundary conditions. However, the methods by these authors are not sufficient to prove that the amplitude functions of a rod form Markov sequences. That the amplitudes form a Markov sequence is the basic property which for the case of orthogonal functions

REMARKS

301

implies the set of oscillatory properties as a simple corollary. In his investigations, S.A. Janczewski found it necessary to classify boundary conditions and to consider many individual cases. The method used here, which is based on the application of Theorem 2 of Section 6 and Rolle’s theorem, yields in a few lines that in all cases considered by Janczewski (when two boundary conditions are applied at each end) as well as in many other cases, the influence function of a rod which corresponds to these conditions is an oscillatory kernel. This method can be also used to prove the oscillatory character of the Green functions of oscillatory differential operators of any order (introduced by M.G. Krein) under various boundary conditions. In particular, this method leads in a very simple way to the results proved almost simultaneously by P.D. Kalafati [25b], by using relatively complicated analytical derivations (for his further research, see [25c]). 3. Theorem 5 of Section 8 is a special case of a theorem by M.G. Krein and G.M. Finkelshtein [31] on Fredholm symbols of oscillatory Green’s functions of differential operators. Chapter IV 1. The treatment of problems of oscillations of an elastic continuum using the theory of loaded integral equations is well-known; nevertheless it is plausible that it appears for the first time in general form with proper explanation and emphasis all relevant mathematical factors in this book. 2. The definition of an oscillatory kernel given in Section 2 (which is in full agreement with the new definition of an oscillatory influence function in Section 5, Chapter III) differs from that definition used in the first edition of the book. The new definition contains additional requirements for the case that one or both ends of the interval are movable. For the case of the influence function and the Green function in general, these definitions are equivalent, as follows from [31]. In the general case, however, the problem of equivalence of the old and new definitions remains open. This new definition of the oscillatory kernel was found to be necessary to develop a complete theory of loaded integral equations, as given in Section 4. 3. In Section 3 we expound on Kellogg’s theory of integral equations with his kernels (except that Kellogg’s uses dσ(x) = dx). The theory is supplemented here only in this respect: unlike in Kellogg’s work, the behavior of the fundamental functions is considered not only within the principal interval, but at all “movable” points. 4. The theory of integral equations with oscillatory kernel and arbitrary distribution function σ(x), as presented in Section 4, has been developed by the authors of this book and is published here for the first time. 5. In Sections 5–10, we present essentially the results of M.G. Krein. We make several remarks. a) Theorem 8 of Section 5 for the fundamental tone was conjectured by Van den Dungen (Cours de technique des vibrations, Fasc. 2, p. 47) for the case that the rod has a constant section and not only internal but also external supports are hinged. In general form this statement was formulated by K. Hohenemser [21]. However, his naive proof, which contains nothing non-trivial, is by no means convincing. A correct proof of the theorem for the fundamental tone was given by M.G. Krein and Ya. L. Nudel’man [32]. In Section 5 we give a simpler proof, which makes it

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possible to establish the result in the most general form (for the fundamental tone and an overtone). b) The main contents of Section 6 are taken from [30b]. In that paper one can also find a theorem on the nodes of forced oscillation of a rod under the influence of a pair of forces with pulsating moment which is applied to a hinge-supported end of a rod. This theorem is obtained as a consequences of several general theorems on the fundamental functions and on the resolvent of an oscillatory kernel which has continuous first derivatives and continuous second mixed derivatives (see [30h]). c) Just as normal Jacobi matrices are the predecessors of the oscillatory matrices, single-pair oscillatory kernels should be considered the predecessors of arbitrary symmetric oscillatory kernels (particularly those which are the Green functions of differential operators). The theory of single-pair oscillatory kernels was first presented by M.G. Krein in his talk at the Second All-Union Congress of Mathematicians [30a], and it was developed on the principles which are indicated in the end of Section 9, independent of the general theory of oscillatory kernels, which was still unknown to the authors at that time. Formulas (165) and (166) of Section 9 for the Fredholm determinant and minor of a single-pair kernel were later generalized by the authors [16d] to include n-pair kernels. In the derivation of formula (164) for the resolvent of a single-pair kernel, we have used the method of this paper. d) Section 10 develops the first two sections of [30e]. Theorem 16 is published here for the first time. Its proof is closely related with more general methods which were developed in the papers [30m,n]. This theorem can be generalized using these methods to the case of an n-pair symmetric Green function, the resolvent of which is an oscillatory kernel for certain negative values of the parameter λ. e) With the aid of the theory of even and odd Kellogg kernels, it is possible to investigate boundary value problems for Sturm–Liouville equations. Also for non-Sturm boundary conditions (i.e., linear boundary conditions, of which at least one connects the values of the function or its derivative at different ends of the interval, rather then at the same end.) A symmetric or non-symmetric continuous kernel K(x, s)(a ≤ x, s ≤ b) is called an even (odd) Kellogg kernel if the conditions   x 1 x2 . . . xn 1◦ K   ≥0 s1 s2 . . . sn x1 < x 2 < · · · < x n   a< 0 2◦ K x1 x2 . . . xn are satisfied only for even (odd) values of n. M.G. Krein has noted that a Sturm–Liouville boundary value problem with periodic boundary conditions   d dy p + qy + λρy = 0, dx dx   dy  dy  = p  , y|x=a = y|x=b ; p  dx x=a dx x=b for all sufficiently small values of λ, leads to odd Kellogg kernels. On the other hand, if the periodicity conditions are replaced by the semi-periodicity conditions:   dy  dy  y|x=a = −y|x=b ; p  = −p  , dx dx x=a

x=b

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303

then the resolvent, for all sufficiently small λ, will be an even Kellogg kernel. Subsequently P.D. Kalafati, using a subtle argument, exhaustively clarified which self-adjoint or non self-adjoint boundary conditions should be imposed in order that for the corresponding resolvent be an even or an odd Kellogg kernel for certain values of λ. The Birkhoff oscillation theorems (Trans. Amer. Math. Soc., 10 (1909)) for second-order boundary value problems with self-adjoint non-Sturm boundary conditions are rather special corollaries of the results of P.D. Kalafati (see his thesis [25b] and the paper “On the Theory of Green’s Functions of Linear Differential Systems of Second Order,” submitted1 for publication). Chapter V Except two theorems of Fekete and Schoenberg, all results of Chapter V belong to the authors. In the first edition of this book they were presented in Chapter II and in the two supplements. Supplement I For the case of symmetric matrices with positive elements, the rule of approximating of the largest eigenvalue (from above and from below) presented in this supplement has been apparently known for a long time. For arbitrary matrices with non-negative elements, it was established by L. Collatz (Math. Zeitschrift, 48, 2). All remaining results of the supplement were obtained for the purpose of generalizing this rule. Supplement II In this supplement are presented several results of M.G. Krein on the mechanical interpretation and application of Stieltjes’ research on continued fractions of a special type which occur in the problem of moments. Concerning the application of continued fractions to the investigation of oscillations, see the works by V.P. Terskikh [56a,b], and by F.M. Dimentberg [10]. On an Application of the Determinantal Inequality for Totally Non-Negative Matrices The paper [16a], in which the authors first obtained the Determinantal Inequality (76) of Section 6, Chapter II, contains the following application of the inequality to the theory of integral equations (for the case dσ = ds)  b ϕ(x) = λ K(x, s)ϕ(s)dσ(s), (∗) a

with totally non-negative, generally speaking, non-symmetric kernel K(x, s) (a ≤ x, s ≤ b). If ∞  D(λ) = cn (−λ)n (c0 = 1) n=0 1 P.D. Kalafati, To the theory of Green’s functions of linear differential systems of 2-nd order, Zapiski Mat. Otd. Fiz.-Mat. Fac. i Kharkov Mat. Obshch. (4) 24, 129-138 (1937) Editor of this translation could not locate this paper in 2001

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is the Fredholm determinant of this equation, then by virtue of (76) of Section 6, Chapter II, we have     b 1 b s 1 s2 . . . sn 0 ≤ cn = dσ(s1 ) . . . dσ(sn ) ··· K s1 s2 . . . sn n! a a  b  b 1 An ≤ ··· K(s1 , s1 ) . . . K(sn , sn )dσ(s1 ) . . . dσ(sn ) = 1 , n! a n! a where

 A1 =

b

K(s, s)dσ(s)

(n = 1, 2, . . . ),

a

from which we obtain that for every complex λ |D(λ)| ≤

∞  An1 |λ|n = eA1 |λ| . n! n=0

(∗∗)

This result can be supplemented by an additional remark. Since equation (∗) can be interpreted as a limit of a system of linear algebraic equations with totally non-negative matrices, one can derive from Theorem 6 of Chapter II that all eigenvalues of equation (∗) are positive. We denote them by λ0 ≤ λ1 ≤ λ2 ≤ . . .  Inequality (∗∗) implies the convergence of the series ∞ i=0 (1/λi ). On the Perron Theorem and Its Integral Analogues As indicated in Section 2 of Chapter IV, Theorem 3 can be extended to the case of non-symmetric kernels, and it is precisely this circumstance that makes it possible to generalize all the main results of Section 3 to the case of everywhere loaded integral equations with non-symmetric Kellogg kernels (about this see the paper by F.R. Gantmakher [15]). Thus, the generalized theorem of Section 3 of Chapter IV should be considered as one of the integral analogues of the Perron algebraic theorem. Various analogues are possible here, depending on whether one considers ordinary or loaded integral equations and to what class the kernel of the equation belongs in the sense of integrability, continuity, etc. The first simple integral analogue of Perron’s theorem was found by R. Jentzsch [23]. The Perron theorem obviously holds not only for matrices with positive elements, but also for matrices A with non-negative elements, if some power of A has positive elements. This fact corresponds to its own integral analogues; on this subject see the paper by M.G. Krein and M.A. Rutman [33] where further literature is given; among the latest papers on this problem is that of T.A. Sarymsakov [47]. It is interesting to observe that the Perron theorem and its integral analogues have a topological proof based on the Principle of Fixed Points [35]. This was shown for Perron’s theorem in the well-known itself book by P.S. Alexandrov and H. Hopf [1]. A topological proof of its integral analogues was found by M.A. Rutman [46], who simultaneously obtained the first generalization of Perron’s theorem to the case of operator equations in Banach spaces. Further investigations in this direction can be found in [33], which contains also detailed literature on this problem and related questions.

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References 1. P. Alexandroff, H. Hopf, Topologie, Springer-Verlag, Berlin, 1935. 2. S. Banach, Th´ eorie des op´ erations lin´ eaires, Chelsea, New York, 1955; English transl., North Holland, Amsterdam, 1987. 3. S. A. Bernshtein, A new method of finding the frequencies of oscillations of ellastic systems (part I), Voenno-Inzhenernaya Akademiya RKKA, Moscow, 1939. (Russian) 4. S. N. Bernstein, Extremal properties of polynomials, ONTI, Moscow, 1937. (Russian) 5a. G. Biezeno and R. Grammel, Die Eigenschaften der Determinanten aus Maxwel’schen Einflusszahlen und ihre Anwendung bei Eigenwertproblem, Ingenieur-Archiv 8 (1937), 364–372. 5b. G. Biezeno and R. Grammel, Technische Dynamik (2 vols.), Springer-Verlag, Berlin, 1953. 6. O. Bottema, Die Schwingungen eines zusammengesetzten Pendels, Jahrsber. deutsch. math. Verein. 42 (1932), no. 1–4. 7a. M. Bˆ ocher, Lecons sur les m´ ethodes de Sturm dans la th´ eorie des ´ equations diff´ erentielles lin´ aires et leurs d´ eveloppements modernes, Gauthier-Villars, Paris, 1917. 7b. M. Bˆ ocher, The published and unpublished works of Charles Sturm on algebraic and differential equations, Bull. Amer. Math. Soc. 18 (1911), no. 1, 40-51. 8. R. Courant and D. Hilbert, Methods of mathematical physics, vol. I, Wiley, New York, 1989. 9a. O. Davidoglou, Sur l’´ equation des vibrations transversales des verges ´ elastiques, Annales ´ Sci. Ecole Norm. Super. (3) 17 (1900), 359–444. ´ ´ 9b. O. Davidoglou, Etude de l’´ equation diff´ erentielle (θy  ) = Kϕ(x)y, Annales Sci. Ecole Norm. Super. (3) 22 (1905), 539–565. 10. F. M. Dimentberg, A method of “dynamical rigidity” with application to determination of f requencies of oscillations of systems with resistance, Izvestiya Akad. Nauk SSSR, Otd. tekhn. nauk 10 (1948), 1577–1597. 11a. N. A. Dmitriev and E. B. Dynkin, On characteristic numbers of stochastic matrices, Dokl. Akad. Nauk SSSR 49 (1945), 159–162. (Russian) 11b. N. A. Dmitriev and E. B. Dynkin, Characteristic roots of stochastic matrices, Izvestiya Akad. Nauk SSSR, Ser. mat. 10 (1946), 167–194. (Russian) 12. M. K. Fage, An extension of Hadamard’s inequality for determinants, Dokl. Akad. Nauk SSSR 54 (1946), no. 9. (Russian) ¨ 13. M. Fekete, Uber ein Problem von Laguerre, Rendiconti Circ. Mat. Palermo 34 (1912), 89– 100. ¨ 14a. G. Frobenius, Uber Matrizen aus positiven Elementen, Sitzungsberichte der Preuss. Akad. Wiss., Phys.-math. Kl. (1908), 471–476; (1909), 514–518. ¨ 14b. G. Frobenius, Uber Matrizen aus nicht negativen Elementen, Sitzungsberichte der Preuss. Akad. Wiss., Phys.-math. Kl. (1912), 456–476. 15. F. R. Gantmakher, On non-symmetric Kellogg kernels, Dokl. Acad. Nauk. SSSR 1 (10) (1936), 3–5. 16a. F. R. Gantmakher and M. G. Krein, On a special class of determinants related to Kellogg integral kernels, Mat. Sbornik 40 (1933), 501–508. 16b. F. R. Gantmakher and M. G. Krein, Sur les matrices oscillatoires, Comptes Rendus Acad. Sci. Paris 201 (1935), 577–579. 16c. F. R. Gantmakher and M. G. Krein, Sur les matrices oscillatoires et compl` etement non n´ egatives, Compos. Math. 4 (1937), 445–476. 16d. F. R. Gantmakher and M. G. Krein, On integral kernels of the Green function type, Trudy Odesskogo Gos. Universiteta, Mat. 1 (1935), 39–50. (Russian)

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Index u-line, 70

forced oscillation, 175 Fourier, 173 Fredholm determinant, 168 minor, 169 symbol, 178 frequency, 116 Fundamental Determinantal Inequality, 91 fundamental frequency, 117 fundamental function, 168

amplitude, 115 antinode, 137 beam, 129 Bernoulli, D., 119 Birkhoff, 303 C-orthogonal vectors, 44 C-orthonormal vectors, 45 Cauchy function, 243 characteristic determinant, 16 characteristic equation, 16, 49 generalized, 42 Chebyshev system, 6, 137, 181, 194 Chebyshev–Hermite polynomials, 128 Chebyshev–Posse polynomials, 128 Collatz, 303 complete system of eigenvectors, 18 compound pendulum, 127 constraint, 117 continuum, 131 segmental, 142

Green function, 237 Hadamard’s inequality, 36, 38, 39, 135 harmonic oscillation, 115, 117 heat kernel, 140 inertia, 117 influence coefficients, 2, 130 function, 2, 130, 131 integral equation loaded, 168 Jacobi formula, 32 matrix, 67 Janczewski, 300 Jentzsch, 304

d’Alembert, 119 Davidoglou, 300 defect, 16 density of a minor, 259 Descartes Rule, 28 diametral hyperplane, 27 differential operator, 236 Dimentberg, 303 discriminant, 24

Kalafati, 303 Kellogg, 180 Kellogg kernel, 180 even or odd, 302 kernel oscillatory, 3, 178, 179 single-pair, 220 Kotelyanskii, 299 Kronecker theorem, 65 Kroneker identity, 32

eigenvalue, 49 of a boundary value problem, 237 of a matrix, 16 of a pencil, 49 of an integral equation, 168 eigenvector, 16 energy kinetic, 113 potential, 113 Euler, 119 exponent of an oscillatory matrix, 76

Lagrange, 119 equations, 114 Laguerre polynomials, 128 law of inertia, 29 loaded integral equation, 3

Fekete, 259, 261, 303 309

310

Markov sequence, 6, 181, 194 system, 7, 258 matrix, 9 adjoint, 12 associated, 64 fundamental, 18 generalized Vandermonde, 76 normal Jacobi, 70 of class d+ , 245 orthgonal, 25 oscillatory, 3 sign-definite, 245 sign-regular, 75 similar, 60 single-pair, 78 strictly sign-definite, 245 strictly sign-regular, 75 symmetric, 19 totally non-negative, 74 totally non-negative of class d, 245 totally positive, 74 totally positive of class d, 245 transposed, 19 triangular, 61 with simple structure, 18 maximin, 53 Maxwell reciprocity principle, 132 Mercer expansion, 170 Mersenne’s Law, 123 minimax, 53, 171 minor almost principal, 91 quasi-principal, 99 movable point, 131, 142 natural oscillations, 117 nodal place, 148 node, 2 of u-line, 70 of a function, 137 of a vector, 70 normal coordinates, 114 number of sign changes exact, 87 maximal, 86 minimal, 86 ocsillate in the same way, 254 oscillatory influence function, 143, 146 matrix, 76 properties, 1, 177 overtone, 117 pencil of forms, 49 phase, 115 Picard, 300 place

INDEX

nodal, 148 zero, 148 Poincar´ e, 173 Poisson, 173 positive pair, 284 principal vector, 27, 49 Principle of Fixed Points, 304 property D + , 248 D − , 248 T + , 246 T − , 246 quadratic form, 23 non-negative, 34, 35 normal Jacobi, 118 positive, 34, 35 rank of, 25 singular, 24 quasi-resonance, 177 resolvent, 169 mechanical interpretation, 176 resonance, 176 Routh, 299 Schoenberg, 253, 303 secular determinant, 1 equation, 1, 16 segmental continuum, 1 shaft, 124 single-pair kernel, 6 singular boundary conditions, 237 stability, 115 stiffness, 117 Sturm, 1, 4, 5, 299 sequence, 105 system, 5, 118 Sturm’s rule, 68 Sturm–Liouville, 7 boundary value problem, 235 support, 133 Sylvester’s formula, 30 identity, 12, 13 symmetry condition, 283 Terskikh, 303 theorem Binet–Cauchy, 10 Mercer, 170 Perron, 83, 182 thread with beads, 119 Volterra equation, 225 zero place, 148