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EXTENSION OF NORMAL THEORY TO GENERAL MATRICES
by Robert I* Lambert
A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject:
Mathematics
Approved:
.. .: : '\ *'’*’• In Charge of Major Work' V " "
Head of
.7 ;~
»r Department
/ f 3 ~FP f Doan of Graduate College Iowa State College 1951
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UMI Number: DP11918
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*li
TABLE Of CONTENTS
I.
Pago INTRODUCTION.................................... 1
II.
PROPERTIES OF NORMAL MATRICES ANDREDUCTION OE A J1ATREC TO NORMAL F O R M ................3
III.
THE NORMAL FORM AND p ( A ) ........................ 6
IT.
THE ALGEBRA OF REAL POLYNOMIALS IN A REAL M A T R I X ...............................14 A. B. C. D. E* F* G.
Automorphisms of R ( A ) .................. Elements of R(A) Which arc Similar to A. Star (Primo) Commutativity with a Positive Definite Herraitian Matrix, The Algebra H ( A ) ........................ A Basis for R(A) and H ( A ) .............. An Idempotent Basis for H ( A ) ........... Isomorphisms of R(A) and R ( A ’) ..........
14 15 18 SO 21 25 27
V.
CONSTRUCTION OF p(A) IF AW IDEMPOTENT BASIS FOR E(A) IS KNOV/N . ...................... 35
VI.
CONSTRUCTION OF AN IDEMPOTENT BASIS FOR H(A) . 37 A. B. C. D. E.
General Problem of Constructing a Basis. 37 A Solution for Rank One Idempotonts . . 38 A General Method for Obtaining Idempotents ............. . . . . . 39 Properties of the S i ’s .................. 44 Numerical E x a m p l e .................... . 4 5
VIIk BIBLIOGRAPHY....................... VIII.
ACKNOWLEDGMENT........................... *
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* . .48 . . 51
1
I.
INTRODUCTION
The problem discussed in this thesis arose from an examination of the methods used to compute the complex characteristic roots of a matrix.
It occurred to the
author that the relative simplicity of the computation of the characteristic roots in the case of a normal matrix indicates that some suitable extension of the normal matrix theory may apply to general matrices. By a general matrix is meant an arbitrary real matrix A with distinct nonzero characteristic roots.
The theory
developed here breaks down for the complex case and the repeated root case and is incomplete for the singular case. A special reduction of a general matrix to a normal matrix leads to a polynomial p(A) which has properties analogous to those of the conjugate transpose of a normal matrix.
The reduction of the general matrix can be made
without specific knowledge of the characteristic roots, but the principal aim of this investigation is to study the properties of p(A) and the application of those prop erties to the characteristic root problem. The philosophy behind the application of this theory to the characteristic root problem is to carry out the
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2analysis in the real number field. be done in linear fashion.
Clearly, this cannot
However, the methods given
here bring the nonlinear aspects of this particular theory within the scope of practical solution. It is helpful to know the nature of the roots of A, that is, the number of complex roots.
This is easily
accomplished by the following well known criterion: The number of distinct real characteristic roots of A is equal to the signature of the matrix
('
S1
s2
Si
Ss
s3
SS
S3
s4
•
•
•
♦
\L
,i Sn
Sji+x .
where si is the trace of A* ,1 The results from Theorem 2.6 on are apparently ariginal although some of the results of Chapter IV are implicit in the general structure theory of algebras. In regard to notation, A ’ is the transpose of A, —. —i A* = A* is the conjugate transpose of A, and x is a vector (x1 ,Xs ,X3 ,...,Xn) .
10, Perron, and Co. 1933.
Algebra, vol. 2, p.2 .
Walter de Gruyter
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ta
il.
PROPERTIUS OP NORMAL MATRICES AND REDUCTION OP A MATRIX TO NORMAL PORM
The definition and some special properties of normal matrices are included in this chapter.
In addition it is
proved that a matrix with linear elementary divisors can be reduced to a normal matrix by a positive definite hermitian similarity transformation. pose of this chapter is twofold:
Specifically, the pur*to prove Theorem 2,6,
and ultimately to make clear the analogy between A? when A is normal and the polynomial p(A) of Chapter III when A is general. Definition 3.1.
A matrix A is normal if and only if
AA* = A*A, Theorem 2.1.
A necessary and sufficient condition
that UAU* = D, where U is unitary and D is diagonal, is that A be normal.1 Theorem 2.2,
If A is normal, and UAU* = D is diagonal
with U unitary, then U(At£*)u* =J5( (a22“an ) must be satisfied.
Note that this condition implies that
agi f 0 as was required above.
This is the precise condi
tion that A have complex roots.
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13The results of this example suggest, in the general case, that p(A) 5 A when A has real roots«
If A has any
complex roots, then p(A) is of degree n1 or less.
Both
of these facts can he easily proved by use of the inter polation formula.
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14
IV.
THE ALGEBRA OE REAL POLYNOMIALS IN A REAL MATRIX
The class of all real scalar polynomials in a given real matrix constitutes a matric algebra over the field of reals.
This algebra will be denoted by R(A).
The
degree of the polynomials in R{A) can be assumed to be less than or equal to u1, where u is the degree of the minimum function of A.
It will be assumed throughout
this chapter that A has distinct characteristic roots. The study of the polynomial p(A) of the previous chapter leads to the study of a special aspect of R(A), namely, that R(A) contains a subalgebra H(A) which consists of all matrices in R(A) with real characteristic roots, so if the matrix A has real characteristic roots then H(A) = R(A).
However, if A has any pairs of complex
roots then H(A) is a proper subalgebra of R(A),
A
detailed analysis is made of possible bases for H(A). There will also be derived some relations between the algebras R(A) and R(A'). A,
Automorphisms of R(A)
Associated with the algebra R(A) is the group of automorphisms of R(A).
Let cr[R(A)] denote the image of
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15R(A) under an automorphism Cr,
All these automorphisms
of R(A) can be found by studying the images of A itself. Assuming that A has minimum equation X4 )
\r1 *2
0
0
0
0
0
1
Cl
0
1
Cl
Cl + Cg
1
Cl
Cl + Cg Cx3+2ciCg / +C3 /
= X4
2 *3
2
x4//
For brevity these equations will be designated by x Si x 1 = xi
,
(i = 1>2 ,3,4).
(48’)
It is to be noted that the matrix S2 is made up of the first column of I, the first column of M, the first 2
3
column of M , and the first column of M .
The matrix
Ss is made up of the corresponding second columns, S3 of the third columns and S4 the fourth columns* The linear term on the right hand side of these four equations can be removed by the substitution x2 = yi+r» x 2 “ 7 2 t x 3 = y 3> x4 = y4 *
Denote the new vector by
y = (yi,7£,y3,y4). The equations (48) become fi
=y
y* = \
fg
a y s8 y' = 0 ^ i*3 “ y S 3 y* = 0
f4
= y S4 y» = 0
(49) .
The four quadratic equations (49) can be solved simultaneously for all of the sets of values of the y i ’s. After finding each set of y i ’s, the corresponding x i ’s are found and hence the corresponding idempotent matrix F.
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42If it is desired to obtain only rank two idempotents from the Equations (49), one need only to require the trace of F to be equal to two.
That is
trF = 4x 1+ c 1x s +( c 12 +2c s )x 3+( c 13+3 c i C2+3 c 3)x 4 = 2. Upon changing to the variables y±, one obtains the homo geneous linear equation 4yi+c1y2+(c12+2c2)y3+(c13+3c1c2+3c3)y4 » 0. (50) Equation (50' can be used in conjunction with Equations (49) to select only those solutions of (49) which will yield a rank two idempotent. Newton’s method of iteration is suggested to solve the system (49).
It will be shown that this iteration always
converges in the general case if the initial values of the variables are sufficiently close to the solution. It is of interest to note that the matrix M prime commutes with each of the matrices Si of Equations (48), that is
MSi = SiM*
It will be shown that M prime commutes with each Si in the general case. It will now be proved that the results of this example axe true in general. n Let F = XiM3**1 i=l n and .
(51) (52)
Then the function F& ~ F becomes Y = F02   l . (53)
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 43
Gonsider the functions
11  y Sj y« (54)
fi ■ y ®i y*
(i * 2,3,... ,n)
where the Si and y are defined as before. The differential of Y in Equation (53) is dY * 3F0dF0  2dF0 *F0
(55)
while the differential of (fi,fg,•..,fn) from (54) is d(fi,fa ,fg,...,fn) i • • «) " 2(dy) (Say ^ S g y ^ S a y S ny»).
(56)
The matrix (S1y*,Ssy ,,.,.,Sny t) « F0 by (58) and the defini tion of the Si. fi « 0,
Furthermore, at y * (&,0,0,0), Y • 0 and s
(1*1,2,.••,n)•
The first row of Equation (55) is the first row of
and hence is the same as (56)*
Therefore the functions (54)
are the same as the functions in the top row of (53), The basic condition for applying the Newton method of iteration1 to the equations fl « .y Sa y«  i  0 4 fi * y Si yt « 0 (i=2,3
n)
requires the nonvanishing of the functional determinant, which in this case is 2F0 , at the solutions.
At the solu
tions, the functional determinant is equal to 2F^lj where
Ostrowski, A, gives an excellent exposition and improve ment of the Newton method in Konvergenzdiskussion und Fehlerabschatzung fur die Newton*sche Methode bei Gleichungssystemen* Commentarii Mathematioi Helvotici. Vol. 9, p. 7983. 19361937.
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443P is the required idempotent, and since F has characteristic roots which are l*s and 0*s, the determinant is nonzero. D,
Properties of the S i ’s, MSi = SiM*
Theorem 6,1, Proof:
,
(57)
The equation MSi = SiM1 is equivalent to the
kronecker product equation (MxM1)8 i » = Si* where
(58)
, ei MSi
Si t a
, ei* is the ith column of I* \ ^ 1 ' ) and Mx M
l
0
M” 1
0
0
0
M~
0
0
0
•
«
•
•
•
•
.
.
0
\
M"1
CnM 1 Cji i M 1 CngM
• • • c^M 1
Direct multiplication verifies the theorem* The number of independent solutions of (58) is exactly n because M x M**1 has exactly n characteristic roots equal
to one.
Therefore the matrices S ^ S g , ,,. ,Sn , which are
obviously linearly independent, are a basis for all solu tions of (57).
In fact the following theorem holds:
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45Theorem 6.2.
Let A = OMO,1? Q, real, and let Ti = QJSiQ,1,
Then the real solutions of the matric equation AX » XA» are the linear set Ti,Tg,...,Tn . Proof:
MSi = SiM’ so that
Hence
ATi ® TiA’ ,
E.
Numerical Example
Consider the matrix ( 0
1
0
0
0
1
0
0
0
0
1
\ 3
5
4
1 )
0 ^
which has one pair of complex roots.
The second and third
powers of M are 0 0
0
1
0
0
0
1
5
5
4
1
\3
8
9
5
(° 3
0
0
1
5
4
1
3
8
9
5
\ 15 28 28 14 From Equation (44), one obtains the polynomial (x2+l)4(x2+l)34(xs+l)25(x2+l)3 * 0,
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4 6 
which has the real roots x2 = 2 and x2 = 2.
Corresponding
to these two solutions are *1 « 1
*1 ■ 1
x2 » 2
x 3 = 2
= 2
*3 a 2
« 1
X4 = 3
Therefore M 3+2MS+2M+I
Qi and
G2 = M 32M82M3I
are rank one matrices. /
Thus the matrices 1
2
3
Gi 1 G°i = trGh “ 52 ,
6
9 \
and
*02
1 4
1
\
6 3
18 18 9
27 54 54 27
/ 3 G2 tr Gs
2
2
2
1 \
3
2
2 1
3
2
2 1
\
3
are rank one idempotents.
2
2
The rank two idempotent F can
be found from F = IGqi—G02 1 /12 52 36
28
28
12
72
20
16
48
4.4
8
4
12
28
28
12
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47The matrix M+p(M) » aF + biGoi + bgGos
3y Equations
(40) and (41), one obtains a * 1 bj » 6 bg ** —2 « Therefore 1 p (M)  — 5M33M216M21l] * JL " 13
21
16
—3
15
4
4
5 \ 2
6
25
12
6 .
18
36
49
18 /
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YU.
BIBLIOGRAPHY
Albert, A.A. Structure of algebras. Pub. vol. 24, p. 151.
A.M.S. Colloquium
Barankin, E.W. Bounds on characteristic values. Bull. Amer. Math. Soc. vol. 54, p. 728735. 1948.


Brauer, A. On the characteristic equations of certain matrices. Bull. Amer. Math. Soc. vol. 53, p. 605607. 1947. Limits for the characteristic roots of a matrix. Duke Math. J. vol. 13, p.387395. Limits for the characteristic roots of a matrix. Duke Math, J. vol. 15, p. 871877.
1946, 1948.
Coddington, E.A. Note on the spectral representation of a bounded normal matrix. Bull. Amer. Math. Soc. vol. 54, p. 736739. 1948. Fettis, H. E. A method for obtaining the characteristic equation of a matrix and computing associated modal columns. Quart. Appl. Math. vol. 8, p.206212.1950, Foulkes, H»0. Collineatory transformation of a square matrix into its transpose. J, London Math. Soc, vol. 17, p. 7080. 1942. Franklin, P. Algebraic matric equations. vol. 10, p. 289314. 1931.
I. Math, Physics,
Leavitt, W.G. A normal form for matrices whose elements are holomorphic functions. Duke Math, J. vol. 15, p. 463472. 1948, Lee, H.C. On the factorization of orthogonal transforma tions into symmetries.Bull. Amer.Math,Soc.vol. 54, p. 558559. 1948. MacDuffee, C.C. The theory of matrices. Publishing Co, p. 76. 1946.
N.Y. Chelsea
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
49” Oldenburger, R, Infinite powers of matrices and char acteristics, Duke Math, J. vol.6,p.357361, 1940, Ostrowski, A, Konvergenzdiskussion und Fehlerabschatsung fur die Newton'sche Methode bei Gleichungssystemen. Commentarii Mathematici Helvetic!,vol.9,p.7983.19361937, Parker, W.V. The characteristic roots of matrices. Duke Math, J, vol. 12, p.519526. 1945. Characteristic roots and the field of values of a matrix. Duke Math. J1. vol. 15, p.439442. 1948. — Limits to the characteristic roots of a matrix. Duke Math. I. vol. 10, p. 479482. 1943.  Characteristic roots of a matrix. vol. 3, p. 484487. 1937.
Duke Math. J.
 Sets of complex numbers associated with a matrix. Duke Math. J. vol. 15, p. 711715. 1948. Perron, 0, p. 2.
Algebra. 1933.
Walter de Gruyter and Co.
vol. 2,
Smiley, M.F. The rational canonical form of a matrix. Bull. Amer. Math, Soc. vol. 48, p,451454. 1942. Taber, H. On a theorem of Sylvester’s relating to non degenerate matrices. Froc. Amer. Acad. Arts Sci. vol. 27, p. 4656. 18911892. Taussky, 0. rices.
Bounds for the characteristic roots of mat Duke Math. J. vol.15, p.10431044. 1948.
Taylor, A.E. A geometric theorem and its application to biorthonormal systems. Bull. Amer. Math. Soc. vol. 53, p. 614616. 1947, Von Neumann, J, and Goldstine, H.H. Numerical inverting of matrices of high order. Bull. Amer. Math. Soc. vol. 53, p. 10211099. 1947. Weyl, H. Generalized Riemann matrices. vol. 3, p. 709745. 1936.
Ann. of Math,
Wiegmann, N.A. Normal products of matrices. vol. 15, p. 633638. 1948.
Duke Math,J.
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50
Williamson, J. A generalization of the polar represen tation of nonsingular matrices. Bull. Amer, Math. Soc. vol. 48, p. 451454. 1942, Matrices normal with respect to a hermitian — matrix. Amer. J. Math. vol. 60, p. 355373, 1938. Note on a principal axis transformation for nonhermitian matrices. Bull. Amer, Math. Soc. vol. 45, p. 920922. 1939, .
.
. . The exponential representation of canonical matrices. Amer. J. Math. vol. 61, p.897911. 1939.
Quasiunitary matrices. p. 715725. 1937.
Duke Math. J. vol. 3,
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5 1 
VIII.
ACKNOWLEDGMENT
The author wishes to express his sincere appreciation to Bernard Vinogradc for his valuable assistance and guidance during the preparation of this thesis.
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