AN ABSTRACT OF THE THESIS OF
VISUTDHI UPATISRINGA for the degree of DOCTOR OF PHILOSOPHY
July 29, 1975
in Mathematics presented on
Title: THE RELATION BETWEEN COMPLEX MATRICES
OBTAINED BY COMPOSING SIMILARITY AND
CONJUNCTIVITY
Abstract approved:
Signature redacted for privacy.
C. S. Ballantine
This dissertation is concerned with the problem of determining,
for given two n x n complex matrices A and B, necessary and suf-
ficient conditions on them so that A is
with B (A is similar to
a matrix which is conjunctive with B). In Chapter II, the 2 x 2
complex case is completely solved. In Chapters III and IV, the
n x n complex case is investigated and is only partially solved. The
n x n complex case includes the following classes of matrices:
Hermitian matrices, diagonable matrices, triangular nondiagonal
matrices, unitary matrices, involutory matrices, positive stable
matrices and accretive matrices. We state three sample results
for the n x n complex case.
Sample Result 1. An n x n unitary matrix A is
with an n x n
nonsingular hermitian matrix B if and only if A = A = A-1 and
index A = index B.
Sample Result 2. An n x n idempotent -natrix A of rank r is Ave,
with an n x n complex matrix B if and only if B is conjunctive with
0
[2I
01 e 1r- s a
0
On- r - s
for some s < r.
Sample Result 3. An n x n real diagonal matrix A with positive
distinct eigenvalues is
with an n x n complex matrix B if
and only if B is a product of three positive definite hermitian
matrices.
The Relation Between Complex Matrices
Obtained by Composing
Similarity and Conjunctivity
by
Visutdhi Upatisringa
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Doctor of Philosophy
June 1976
APPROVED:
Signature redacted for privacy.
Professor of Mathematics
in charge of major
Signature redacted for privacy.
Ch
man of Department of Mathematics
Signature redacted for privacy.
Dean of Graduate School
Date thesis is presented
July 29, 1975
Typed by Ilene Anderton and Lyndalu Sikes for Visutdhi Upatisringa
ACKNOWLEDGMENT
I would like to express my sincere gratitude to Professor
C. S. Ballantine for suggesting the problem, for his guidance and
encouragement, and for his willingness to sacrifice so much of his
time and energy throughout the preparation of this thesis.
I would also like to thank Professor Harry E. Goheen for
his initial encouragement of my studying mathematics.
Finally, I would like to thank my wife, Sally, for her patience
and understanding.
TABLE OF CONTENTS
Chapter
I.
II.
INTRODUCTION AND PRELIMINARIES
2 x 2 COMPLEX MATRICES
§2. 1.
§2.2.
III.
Singular Cases
Nonsingular Cases
n x n COMPLEX MATRICES WITH REAL
EIGENVALUES
§3. 1.
§3. 2.
IV.
1±a_se
17
19
35
Diagonable Matrices
64
64
Triangular Matrices
73
n x n COMPLEX MATRICES WITH COMPLEX
EIGENVALUES
§4. 1.
§4. 2.
§4. 3.
§4. 4.
§4. 5.
1
Diagonable Matrices
Triangular Matrices
Unitary Matrices
Positive Stable Matrices
Accretive Matrices
77
77
83
93
96
98
BIBLIOGRAPHY
121
PAGE INDEX TO LEMMAS, THEOREMS, ETC.
123
LIST OF TABLES
Page
Table
2 x 2 singular complex matrices.
61
2 x 2 nonsingular complex matrices.
62
n x n complex matrices.
115
Matrices similar to triangular matrices.
119
Miscellaneous results.
120
THE RELATION BETWEEN COMPLEX MATRICES
OBTAINED BY COMPOSING
SIMILARITY AND CONJUNCTIVITY
I.
INTRODUCTION AND PRELIMINARIES
In this dissertation we are investigating certain relations
between similarity and conjunctivity of n x n matrices with entries
in the complex field C. Our problem is to find necessary and
sufficient conditions on two n x n complex matrices S and T (or on
two classes of such matrices) so that S is similar to a matrix which
is conjunctive with T, or S is conjunctive with a matrix which is
similar to T. In Chapter II, we deal mainly with 2 x 2 complex
matrices. In Chapter III, we restrict ourselves to some n x n
complex matrices with real eigenvalues. In Chapter IV, we look
into some more general n x n complex matrices with complex eigenvalues.
We first recall that two n x n complex matrices S and T are
said to be conjunctive if there exists an n x n nonsingular matrix D
such that D SD = T (where D denotes the conjugate transpose of D),
and S and T are said to be similar if there exists an n x n nonsingular
matrix C such that C-1SC = T. We give the following definitions for
the relations of our problems.
2
Definition 1. 1 An n x n complex matrix S is said to be CAUL to
an n x n complex matrix T if there exist n x n nonsingular matrices
C and D such that D* SD = C-1TC.
Definition 1. 2 An n x n complex matrix S is said to be -1/kle
with an n x n complex matrix T if there exist n x n nonsingular
*
matrices C and D such that C -1 SC = DTD.
We consider a few basic properties of the above relations.
Lemma 1. 1 Let S and T be n x n complex matrices.
Then S
is Ake with T if and only if T is 1t to S.
Proof: The proof follows directly from Definitions 1. 1 and 1. 2.
Lemma 1. 2 Let S and T be n x n complex matrices and g be
a complex number. Then (1) if S is Cdrytto T, then 6S is Gut to ET,
and hence (2) if T is Aisle with S, then ET is 404e with eS,
Proof: Let S and T be n x n complex matrices and fi be a
complex number. Then (1) if S is C64 to T, then there exist nonsingular matrices C and D such that D SD = C - TC.
CC
-1
So &(D *SD)
=
TC) and hence D (6S)D = C1- (ET)C. Thus ES is Owl., to ET.
(2) follows immediately from Lemma 1. 1.
Lemma 1. 3 For every n x n complex matrix S, S is jibe with
S.
Proof: Trivial.
Lemma 1. 4 For every n x n complex matrix T, T is °hut to T.
Proof: Trivial.
3
We can easily show that the relations ).,e and eid, are not
symmetric. This property is illustrated in the following example.
Example 1. 1. Let S =
Here S is
10
[ 2
21
and T
=
r
1
0
01
1'
with T, since S is similar to the matrix [ 1
is conjunctive with T. However, T is not
0]
2
which0
with S, since S is
not positive definite hermitian.
Lemma 1. 5 Let S and T be n x n complex matrices.
if S is conjunctive with T, then S is
Then
with, and eAzi to, T;
if S is similar to T, then S is4i with, and
ei,ti
to, T.
Proof: Let S and T be n x n complex matrices.
(1) Since S
is similar to itself and T is similar to itself, S islose with T, and
S is eivg to T, respectively. (2) Since T is conjunctive with itself
and S is conjunctive with itself, S is,/Ate with T, and S is 6.4 to T,
respectively.
We shall always denote the rank of the matrix S by rank S.
Lemma 1. 6
If an n x n complex matrix S is .,Aie with an n x n
complex matrix T, then rank S = rank T.
Proof: If an n x n complex matrix S is 4 with an n x n com-
plex matrix T, then there exist n x n nonsingular matrices C and D
such that C-1SC = D TD. Since C and D are nonsingular, rank S =
rank C-1SC = rank D TD = rank T.
4
Lemma 1. 7 If an n x n complex matrix T is
to an n x n
complex matrix S, then rank T = rank S.
Proof: The proof follows from Lemmas 1. 1 and 1. 6.
Lemma 1. 8 Let T be an n x n complex matrix with nonzero
determinant and S be an n x n complex matrix. If S is
with T,
then sgn(det S) = sgn(det T).
Proof: Let T be an n x n complex matrix with nonzero deter-
minant and S be an n x n complex matrix. If S is,Afe with T, then
there exist n x n nonsingular matrices C and D such that C-1SC =
D TD. Since det S = det(C- 1SC) = det(D TD) = (det D D)(det T)
and det (D D) >0 for every nonsingular matrix D, we see that det S
is a positive multiple of det T, i. e.
sgn(det S) = sgn(det T).
,
Lemma 1.9 Let S be an n x n complex matrix with nonzero
determinant and T be an n x n complex matrix. If T is Oki to S, then
sgn(det T) = sgn(det S).
Proof: The proof is essentially the same as that of Lemma 1. 8.
We shall follow the notation in [2] for matrix entries and for
submatrices. Let S be an n x n complex matrix and let M and N be
nonempty subsets of the set {1,
2,
..
.
,
n} and let M' and N' be the
subsets complementary to M and N respectively. We denote by
S[ M N] the submatrix of S lying in rows whose indices come from
M and in columns whose indices come from N. We denote by S.. the
entry in the 1 x 1 submatrix S[il j]. We give meanings to three
5
other ways of designating submatrices: S[M
S(
N]
S[ M' I N], and S(M N) = S[
I N' I.
= S[MIN'],
We abbreviate further
for principal submatrices, putting S[ M] = S[M I M] and S(M)
S[
]NT].
Next we say that a nonsingular matrix D defines the
D SD and that the order of this conjunctivity is
conjunctivity S
the order of D. Finally, we let m be the cardinal of M and let Q be
an m x m nonsingular matrix. Then by "the [M] subconjunctivity of
order n defined by Q" we mean the conjunctivity (of order n) defined
by the n x n matrix D satisfying D[M] = Q, D[M1M)
0, DWI Mi
I, where I is the identity matrix of order n-m,
0, and D(M)
(We
shall usually not specify the order of a subconjunctivity when it is
clear from context. ) When i < j, the [1, j] subconjunctivity (of order
n) defined by
r
0l j
L1
will be called "the interchanging [1, j] subconjunctivity (of order n).
The following fact from [ 2], characterizes the effect of an arbitrary
subconjunctivity.
Fact 1, 1, ([ 2], Fact 1, I) Let S be an n x n matrix, M be a
subset of {I,
2,
.
,
n}, m be the cardinal of M, Q be an m x m
nonsingular matrix, and T be the matrix obtained by applying to S
the EM] subconjunctivity (of order n) defined by Q. Then
6
T[M] = QS[M] Q
T[M1M) =Q*S[MIM)
T(MIM] = S(MIM1 Q
T(M)
S(M)
Hence any zero columns of S[M IM) and of T[MIM) correspond, and
any zero rows of S(M IM] and of T(MIM] correspond,
We state two more facts about triangularizing matrices by
conjunctivity,
Fact 1. 2. ([2], Fact 1.2) Let A, B, C, D be matrices of
respective dimensions p x p, p x q, q x p, q x q, and let A be nonsingular. Then there are uniquely defined matrices C and D1 such
J
that the conjunctivity of order p + q defined by the matrix (in block
form)
I
[
0
-1
-AB
,
I
-1
has the following effect:
A
C
B
D
[
A
0
C1
D1
Fact 1.3. ([2], Fact 1.3) Let S be an n x n matrix and let
r be the rank of S. Then S is conjunctive with a lower triangular
matrix whose first r diagonal entries are nonzero and whose last
n - r columns are zero.
7
Next, let S be an n x n complex matrix. We shall define the
"conennS), to be the set of all complex numbers X SX for which X
is an n x 1 complex matrix. We list some basic properties of the
function Fin the following result.
Fact 1.4. ([ 2], Face 3.1). (i) F(zS) = (sgn z) F(S) for every
nonzero z
C; (ii) ris conjunctively invariant, i, e. , [ID SD) =
PIS) for every nonsingular D and every S of the same order as D;
(iii) F(S1)
FI(S) for every principal submatrix S1 of S; (iv) every
diagonal entry of S lies in r(S); (v) r(S) consists of zero, and the
numbers T11 for which T is conjunctive with S (in particular, F(S)
contains all the eigenvalues of S); (vi) r(S) = r(51) + r(S2) whenever
S2 (()denotes the direct sum); (vii) r(S) is a convex cone
S = S1
for every S.
As in [ 2], we introduce some algebraic definitions about S
that will correspond to the various geometric possibilities for the
cone r(S). For each n x n complex matrix S and each real 0 (0 will
always be real when used in this dissertation) we define a (hermitian
n x n) matrix H(O;S) by
H(O;S)
-iOS
+
e0 S*).
(When S is understood, we shall sometimes write H(0) instead of
H(O;S). )
We also use the well-known fact that the only kinds of
cones in C are: the zero cone (the origin); half-line (closed half-line
8
with endpoint at zero); line (line through zero); sector (convex sector
containing zero with apex at zero); half-plane (containing zero with
boundary line through zero); and C itself, An n x n complex matrix
S will be called:
contrahermitian
iff e -i0 S i s nonhermitian for all 0
iff r(S) has nonempty interior;
cohermitian
iff S is not contrahermitian
iff r(S) is a subcone of some line;
contradefinite
iff H(e) is indefinite for all
iff r(S) = C (the whole plane);
codefinite
iff S is not contradefinite
iff r(S) is a subcone of some halfplane;
unidefinite
iff there is a 0 for which e -i0 S is a
nonnegative definite nonzero hermi-
tian matrix
iff r(S) is a half-line;
bidefinite
iff there is a 0 for which e -i0 S is an
indefinite hermitian matrix
iff r(S) is a line;
transdefinite
iff there is exactly one value of ei0 for
which H(0) is nonnegative definite
and nonzero
9
iff r(S) is a half-plane;
iff S is codefinite but not transdefinite
prodefinite
nor cohermitian
iff r(S) is a sector;
iff S. = 0 for all i, j = 1, 2,
the zero matrix
n
iff PS) is the zero cone.
In [ 2] the first of each of the above pairs of "iffs" is given as
the definition of the underlined word,
We next state some properties of the function E defined as in
[ 2].
Let
denote the set of n x n complex matrices all of whose
real eigenvalues are positive.
For S E
,
the function E is defined
as
1
tr{[(1-t)I + tS]-1 (S - I) } dt
E(S) =
0
(where 77
,! means "imaginary part" and "tr" means "trace"). Let
denote the set of n x n nonsingular complex matrices S for which
-1 (if F(S),
The following two facts are from [ 2],
Fact 1. 5.
...
12X
([2], Fact 3. 2) Let S e , and let X ,X
be the eigenvalues of S.
For j =
I.,
2,
<,, ) of arg X.. Then
E(S) =
cr. .
j=1
n
n let o-. be the prinJ
cipal value (1. e.,
'
10
(thus E(S) is a particular determination of arg (det S)). Hence
e
iE (S)
= sgn (det S)
Fact 1.6. ([2], Fact 3.3). E is conjunctively invariant on
E(D SD) = E(S) for every n x n nonsingular D and every
a,',i.e.,
S
E
.
As in [ 2], let 5) denote the set of all n x n positive definite
complex (hermitian) matrices. For each positive integer m let
denote the set of all matrices S such that S can be written as a
product of m matrices from?. We state the following two theorems
of [ 2] without proof.
Theorem 1. 1.
([2], Theorem 2). Let S be an n x n complex
matrix. Then the following four statements are equivalent (to each
other):
S
y--)2
(i. e., S is a product of two positive definite her-.
mitian matrices);
S is similar to an element of (D2
S is similar to an element of p;
s is unitarily similar to a diagonable lower triangular
matrix of positive diagonal.
(Note: "diagonable" means "similar over the complex field to a
diagonal matrix". )
11
Theorem 1. 2 ([ 2], Theorem 3). Let She an n x n complex
matrix. Then the following six statements are equivalent;
SE
93(i. e. , S is a product of three positive definite
hermitian matrices);
S is conjunctive with an element of P3;
S is conjunctive with an element of P2;
S is conjunctive with a lower triangular matrix of positive
diagonal;
S is conjunctive with a matrix all of whose leading principal
minors are positive;
at least one of the following (vi. a) or (vi. b) holds:
(vi. a)
det S> 0 and S is contradefinite;
(vi. b)
S
6. and E(S) = 0 and 1 E r(s).
An extremely useful result of [ a], which we shall use
repeatedly, is the following lemma.
Lemma 1.10. ([ 2], Lemma 4.1).
(i) Let 13 be real and y be
Then there is a conjunctivity having the effect
nonnegative.
r
1
0,
L
2y
1
e
r
eia
L
2(y2
sin2a)k
for all real a such that
sin a < y and (y - sin a)2 2 + COS a > 0
.
12
> 1, and for all a such that I sin al <y and
i. e. , for all «when
cos a> 0 when 0 < y <1.
(ii)
Thus, in particular, whenever 1-Tr
> Ial> 161, every
2
matrix of the form
= eir3
1
r
eia
0
2p
e-ja
is conjunctive with a suitable matrix of the form
io
A2 =
°-i6]
eir3 [ eX
2
(Where p is the same in A2 as in Al), namely, with one for which
IP12 + sin2 a = I12 + sin 2
8.
(Thus K i0 if 12-Tr >
1a
> 161.)
Furthermore, " 1 a = Tr and p 0 in Ai, then Ai is contradefinite and is for every 8 conjunctive with a suitable matrix of the
(iii)
form A2 (namely, with one for which
P12 +
= lx
2
+
sin2 8).
1
In applying Lemma 1. 10, we often use the following notation.
Let T be an n x n nonsingular lower triangular matrix and, for j =
1,
2,
n, let p-
arg T... We then define arg diag T by the
J3
equation
arg diag T = (cr 1,
0-
2'
y
0- n)
We shall write
(0-
1,
0-2,
n)
(1-1,
T2'
n)
13
as an abbreviation for the following conditional: "If T is any
(nonsingular lower triangular) matrix whose arg diag is
o- 2,
( cr
(T1
'
. ,
T'2
CT n), then there is a matrix whose arg diag is
and which is conjunctive with T.
n
Another useful fact mentioned in [2] is the following lemma.
Lemma 1.11. For arbitrary complex numbers a, c, and d,
the two matrices
[a
and
o]
[
d
c
o,
a
-I
are conjunctive.
Proof: We consider the following three cases:
Case 1.
c =0
Case 2.
c
0
1
[1
0
[
ra
0
0
0
d
0
L1
1
0J
=
-
0 ----a
rc
0I
0
0
_T
[ 0
a
aE
C
0
C
a
0
0
1
2
C
c
a
a
0
a
0
0
1
0
--Ed
ca
1
T
I
1
0
1
Ca-E
Case 3. cd
d
0, d = 0
ac
1
a
c
d
c-c-1
-Ed
c
0
al
0
ca
0
a
i
14
We define an n x n complex matrix S to be *-regular (or EP
or EPr of [9]), provided SX = 0 implies S*X = 0 for n x 1 complex
matrices X. We define an n x n complex matrix S to be *-irregular
if it is not *-regular. We state without pi-oof the following known
results.
Lemma 1.12. ([8], Theorem 1. 1).
Every codefinite matrix
is *-regular.
Lemma 1.13. ([8], Theorem 1.2). An n x n matrix S of rank
r is *-regular if and only if S is unitarily similar to the direct sum
of a nonsingular r x r matrix and a zero matrix.
Lemma 1. 14. ([8], Lemma 6.9). If S is n x n lower triangular
nonsingular contrahermitian with all diagonal entries on some line
through 0 and not all on one side of 0, then S is contradefinite, in
fact, S has a 2 x 2 or 3 x 3 principal submatrix which is contradefinite.
Lemma 1. 15. ([8], Lemma 4. 2). Let S be n x n contra-
definite nonsingular and t1,
such that sgn ( t l't 2'tn)
= sgn
tn be any complex numbers
det S. Then S is conjunctive with
a lower triangular matrix T such that diag T =(ti, t2,
T
,
tn) and
n] is contradefinite.
An n x n complex matrix S is said to be trapezoidal if it is
lower triangular and its first r diagonal entries are nonzero (where
r = rank S).
Thus Fact 1. 3 says that every n x n matrix is
15
conjunctive with a trapezoidal matrix.
Lemma 1. 16. ([8], Lemma 3. 10).
Let S be *-irregular n x n
,
of rank r and let t1' t2'
tr be any nonzero complex numbers.
Then S is conjunctive with a matrix T such that
T is trapezoidal,
= 1, and
Tr+1, r
diag T = (t1, t2,
,
tr, 0, 0,
,
0).
Finally, we state without proof the following well-known results
for reference.
Theorem 1.3. ([11], pp. 100-101). Two n x n hermitian
matrices are conjunctive if and only if they have the same rank and
index.
Theorem 1.4. ([10], pp. 84-85). An n x n complex matrix P
is positive definite if and only if P = D D for some n x n nonsingular
matrix D.
An n x n complex matrix U is unitary if U U = I and an n x n
complex matrix S is normal if SS = 55.
Theorem 1.5. ([7], p. 314, p. 316). If S is an n x n hermitian
(or normal) matrix, then there exists a unitary matrix U such that
U
-1 SU is diagonal.
Theorem 1.6. ([11], p. 194). Two n x n normal matrices
are unitarily similar if and only if they have the same eigenvalues.
16
Theorem I. 7. ([10], p. 67). (Schur's Triangularization
Theorem). If S is an n x n complex matrix with eigenvalues X
X 2,
,
X n, then there exists an n x n unitary matrix U such that
U SU = T is a lower triangular matrix with
diag T = (X 1,X 2,
, X n).
Theorem 1.8. ([6],
p.
142; [12], p. 357). (Autonne's
Lemma). Let S be an m x n complex matrix.
Then there exists an
m x m unitary matrix V and an n x n unitary matrix W such that
v sw
A
[ 0
o
0'
where A is a square diagonal matrix all of whose diagonal entries
are positive.
17
2 x 2 COMPLEX MATRICES
II.
In this chapter we consider only 2 x 2 complex matrices. Here
we have determined the necessary and sufficient conditions for two
with, or
2 x 2 complex matrices to be 1/4-t&
e#10.44,
to
each other. First, let
S
Eac
loco
be an arbitrary 2 x 2 complex matrix. We make the following
abbreviations:
det S = ad - bc
A
= k (ad- +
We see that I. <
I
'
-
-
and we can further verify that A and .T.are
invariant under unimodular conjunctivity. The ratios A:
I Al
are thus invariant under arbitrary conjunctivity, as are the
inequalities (I. >0,
> 0, A >0,
<- , etc.
As in 2], we list canonical forms for S under conjunctivity
I
and give criteria (for deciding which canonical matrix S is conjunctive with) in terms of conjunctivity invariants.
the nonsingular cases of S. Here A
First, we consider
0, so we can define a real
number 3 (mod Tr) and a real nonnegative number y satisfying
18
e2ip
161-. 1
1 - 2y 2 = 16 I
-1
Thus 13 (mod Tr) and y are conjunctivity invariants of S.
Case 1.
S is bidefinite. Here S is conjunctive with
il3 r i
01
L0
-i-j
e
in which either determination may be taken for
characterized by the condition:
ei13.
This case is
0 and S* = -e-2ip S.
A
Case 2. S is nonsingular but not bidefinite. Here S is
conjunctive with
ip
e
1
2,/
0
11
where either determination may be taken for
eiP
if S in contradefinite.
When S is codefinite, the determination above is conjunctively
invariant and satisfies
Re
-ip
{e
(a + d) } >0.
Case 2 is characterized by the condition: 6
0 and S*
- e-2.43S.
Next we consider the singular cases.
Case 3. S is singular and contradefinite. Here S is conjunctive with
19
r0
01
12
0-1
Case 3 is characterized by the condition: A = 0 and I> <0.
Case 4. S is singular and codefinite and nonzero. Here S
must be unidefinite and the trace of S must be nonzero. Thus we
can define eip to be the signum of a + d. Then e ip is a conjunctivity
invariant of S and S is conjunctive with
[e0iP
001
This case is characterized by the condition: S
0 and A = 0 and
C.= 0.
Case 5. S = 0.
§2.1.
Singular Cases
Theorem 2.1.
Let A be the 2 x 2 zero matrix and B be a
2 x 2 complex matrix. Then B is
e6/4_,
to A if and only if
B = 0.
Proof: Trivial.
Theorem 2. 2. Let B be the 2 x 2 zero matrix and A be a
2 x 2 complex matrix. Then A is
A = 0.
A-114.&
with B if and only if
20
Proof: Trivial.
Lemma 2.1. A 2 x 2 complex matrix A is similar to a positive
semidefinite hermitian matrix of rank one if and only if det A = 0 and
tr A >0.
Proof: Suppose a 2 x 2 complex matrix A is similar to a
positive semidefinite hermitian matrix H of rank one. Then there
exists a 2 x 2 nonsingular matrix C such that C-1AC = H. Since
-
rank H = 1, det H = 0. Thus det A = det (C 'AC) = det H = 0.
Because H is positive semidefinite nonzero, tr H > 0, so tr A =
tr
(C-1AC) = tr H > 0.
Conversely, suppose A is a 2 x 2 complex matrix with
detA= 0 andtrA> 0. Then the characteristic equation of A is
det (X I - A) = X2 -X (tr A) = O.
Thus the eigenvalues of A are trA and 0. Since these eigenvalues
are distinct, there exists a nonsingular matrix C such that
C-1AC = r
trA 0
-0
01
-1
which is clearly a positive semidefinite hermitian matrix of rank
one.
Since every two 2 x 2 positive semidefinite hermitian matrices
of rank one are conjunctive, we have the following theorem.
21
Theorem 2.3. Let B be a 2 x 2 positive semidefinite hermitian
matrix of rank one and A be a 2 x 2 complex matrix. Then A is
__,.Ziite,
with B if and only if det A = 0 and tr A> 0.
Proof: The proof follows from Lemma 2.1 and the above
remark.
A 2 x 2 codefinite complex matrix B of rank one is conjunctive
(by case 4 at the beginning of this chapter) with
ip
e
where e
1
[0
0
01
sgn (tr B) and is a conjunctivity invariant of B. Thus
we have the following result.
Corollary 2. 3.1. Let B be a 2 x 2 codefinite complex matrix
of rank one and A be a 2 x 2 complex matrix. Then A is _Abi-t6
with B if and only if det A = 0 and sgn (tr A) = sgn (tr B).
Proof: The proof follows from Theorem 2. 3 and the above
remark.
Theorem 2. 4. Let B be a 2 x 2 contrahermitian matrix of
rank one and A be a 2 x 2 complex matrix. Then A is
with B if and only if A has rank one.
22
Proof: Let B be a 2 x 2 contrahermitian matrix of rank one
and A be a 2 x 2 complex matrix. ("Only if") LetA be ;Me. with B.
Since B has rank one, by Lemma 1.6, A has rank one.
("If"). Suppose A has rank one. We consider two cases.
Case (i):
trA = 0. Here A has single eigenvalue 0 and hence is
similar to
0 01
L2 0-1
r
which is conjunctive (by Case 3 at the beginning of this
Chapter) with B. So A is
Case (ii);
trA
0.
_,1114.E
with B.
Here A has eigenvalues trA and 0. So A is
similar to the matrix
rtrA 0
01,
L2
which is conjunctive (by Case 3 at the beginning of this
Chapter) with
00
E
Hence A is
2
0]
with B.
Lemma 2. 2. If a 2 x 2 complex matrix B is conjunctive with
a matrix of rank one and trace zero, then det B = 0 and t.(B) <0.
23
Proof: Suppose a 2 x 2 complex matrix B is conjunctive with
a matrix of rank one and trace zero. Thus B is conjunctive with
with a2 + bc = 0 and not all of a, b, c are zero. Hence there exists
a nonsingular matrix D such that
b
D BD = [ca -al
So det (D*BD) =-a2-bc = 0, and hence det B = 0.
Now,
BD) = k fa (-a) + a (-a) - bb - cc} ,1/2 {2 aa + bb + cc}.
Since 2 aa + bb + cc = 0 forces a = b = c = 0, contradicting our
assumption on a, b and c,
we must have 2 aa + bb + cc > 0. Thus
(D BD) <0. Since the inequality
<0 is conjunctively invariant,
(B) <0,
Theroem 2. 5.
Let A be a 2 x 2 complex matrix of rank one
and trace zero, and let B be a 2 x 2 complex matrix. Then B is
efsIAL,
Proof:
to A if and only if det B = 0 and .T. (B) <0.
Let A be a 2 x 2 complex matrix of rank one and trace
zero, and let B be a 2 x 2 complex matrix.
("Only if").
Let B be
efra,
to A. Then there exist
nonsingular matrices C and D such that D BD = -1
CAC. Since
24
-1
*
rank (D*BD) = rank (C-1AC) = rank A = 1 and tr (D BD) = tr (C AC)
0, by Lemma 2. 2, det B
trA
0 and
(B) <0.
("If "). Suppose det B = 0 and .(B) <0.
Then B is conjunctive
(by Case 3 at the beginning of this Chapter) with the matrix
00
[2 01
which is similar to A. Thus B is
to A.
Theroem 2, 6. Let A be a 2 x 2 complex matrix of rank one
and trace
cm".
0, and let B be a 2 x 2 complex matrix. Then B is
to A if and only if det B = 0 and either
(B) <0 or
sgn (trB) = sgn (trA).
Proof:
trace
Let A be a 2 x 2 complex matrix of rank one and
0, and let B be a 2 x 2 complex matrix.
("Only if"). Let B be
trA
CA4/
to A. Since rank A = 1 and
0, B is conjunctive with a matrix of rank one and trace
0.
Hence det B = 0 and (by conjunctivity canonical matrix at the begin-
ning of this Chapter) B is conjunctive either with
[
ei13
0
2
0
where
0] or with eiP [10 0]
0
= sgn (trB), If B is conjunctive with
r0
01
L2
0-1
25
then 1)(B) <0; and if B is conjunctive with
ip
0
1
01 '
E0
e
then sgn (trB) = sgn (trA).
("If"). Suppose det B = 0 and either ts (B) <0 or sgn (trB) =
sgn (trA). If
(B) <0, then B is conjunctive with the canonical
matrix
00
[2 01
which is again conjunctive with
rtrA 0
01
L2
Since the eigenvalues of A are trA and 0, and trA
rtrA
L2
is similar to A. So B is
Oktt,
0,
00]
to A.
If .T.(B) > 0 and sgn (trB)
= sgn (trA), then B is conjunctive with the canonical matrix
ip
1
[0
e
0
01 '
where eiP = sgn (trB), and hence B is conjunctive with
trA
I- 0
So B is
OPZAL,
to A.
0]0
26
§2. 2.
Nonsingular Cases
We first prove a few lemmas.
0
1
Let B = [ 2.y
Lemma 2. 3.
I with y > 0, and let D = [a
with a, b, c, d EC. Then tr (D BD) =
Proof: Let B =
a
D = [c
D*BD
1
0
2.y
1]
b
=
+c
+ (rd. + 2(aC + bU) y
+ b-S +
with y > 0, and let
with a, b, c,
{-a1-3
d,
E
Then
C.
0,
[1
-az]
2-y
+ 2 aZ- y
-I-
bb + cid + 2 ba y
y.
Hence tr (D BD) = a-at + bE + Cc- +12 + 2 (ac +
Let a, b, c,
Lemma 2.4.
Then
(i)
1
1b12
12
+
lb 12
+ Id 12 > 0;
c
with ad
bc.
0;
12 >0;
la 12
1
dE C
Id 12
1c12
la + c 12 +lb + d
Proof: (i). Suppose
12
> 0.
Ia
2
bc
1
r2
2
+1bl +1c1 +idl
2
= 0.
27
Then we must have a=b=c=d= 0, forcing ad = bc, a contradiction.
(ii) and (iii) can be similarly argued as in (i). (iv). Suppose
+
c2 -I- lb + d I2 = O.
Then we must have la +cl= lb + di= O.
1
Hence a = -c and b = -d, forcing ad = bc, a contradiction.
Let y > 1. Then
Lemma 2. 5.
rl
0
.
1] is conjunctive with
-L
[
1
0,
2y
1
Proof:
o
11
1
- [ 2y
rl
0]
=
[
-1
-1
-2y -10] is conjunctive with
01
[
-1
0
[10 -1]
-1
= [2y
0,
-11
Let y > 1. Then,by Lemma 1.10,
r
-1
L 2y
o.sis conjunc
-1
Lemma 2. 6.
B= e
]
with
1
01
2
1'
Let
L
2y
1with y > O.
Then B is conjunctive with a matrix of zero trace if and only if
,
28
Proof: Let
B=e
1[3 r
0
1
2y 1]
1
with y > O.
We assume without loss of generality that eiP
= 1.
("Only if").
Suppose B is conjunctive with a matrix of zero trace. Then there
exists a nonsingular complex matrix
such that tr (D BD) = 0.
Thus,by Lemma 2.3,
+ dra + 2 (aE + b-a) y = 0. Since by
+c
a-g:
Lemma 2. 4 (i), a + bE + c + d> 0, we must have (aC- + ba) y <0.
Since y > 0, it follows that a7 +15a <0. Now,
0=
+ b17. +
= a-a- +b17; +
=
+ bE +
y
+ (1,71. + 2 (a-c- +
+
+ (a-c- +
+ Ecl ) y
+
+-aic + -b-d) + (y -1) (aT+b-a+T.c +Ed)
+ (fa + (a-c- +
.1a+c12+1b+d12+ (y -1) (a-C- +
Since by Lemma 2.4 (iv),
(y - 1) (iE
a
c
12
+
+ac +13d)
b + dl2 > 0, we must have
+Ecl) <0. Since a-c- + ba +--a.c +Ed <0, it follows
that y - 1 > O. Thus y > 1.
("If").
Let y > 1. Then it suffices to show that there exists
a 2 x 2 nonsingular matrix
29
ab
D = [c
such that tr (D BD) = 0. Let
1
D = [ -(y
-("Y
+ j2
-
Then D is nonsingular since det D = 1 - (y + iy2 -
D4BD
=
(y
1
eiP
/ 2
-( y + VN
1)
[ - (y
+j-y2 -1)
1
ip
a.
(N+iN
[=
=e
1
+A2_0
1 - (.y2 - y2 + 1
eip
r0 *
0. Thus
1 01
2-y
1
-h,
1
+17-12 )][
1)2
(Y+42-1)+
where *'s are numbers of no interest to us.
Hence tr (D BD) = 0.
The selection of the above nonsingular matrix D is motivated
by the remark below. Since
*
DBD
=
ip
r
+ cc' + 2 a-c-
b71; + (fa + 2 ba y
it suffices to find a, b, c, d E C with ad
bc satisfying
30
aa + cc + 2 ac
and
bb
=0
(1)
y=0
dd--+ 2 1371
( 2) .
By Lemma 2. 4 (ii) and (iii), we have a-a- + cc > 0 and bi-; + dc-T> 0.
Since N > 1, we must have aT <0 and ba <0. Thus
a
a_c
=
cc
C
<0 ana
<u.
=
Let
-- = a and -u =
1
with t > 0 and u > 0. Then c = -at and b = -du. Substituting b and c
into equations (1) and (2), we have
Ia
i2
(t - 2 -y t + 1 ) = 0 = Id
2
- 2 Nu + 1).
Thus
t= N
\i/N 2 -
1
and
u=
Since bc = (-du) (-at) = (tu) (ad) and ad
iN 2
bc, it follows that tu
Thus we can take
t=u= -y +
Lemma 2. 7.
2
-1 and a=d=1.
Let
1
0
B = [2y
i]
-1.
1.
31
with 0 < y < 1 and let Q be any nonsingular 2 x 2 matrix. Then
Re {tr (Q*BQ)) > 0.
Hence if A is j141.1.
with B, then Re {trA} > 0.
Proof: Let
1
B = [2y
0
1]
with 0 < y <1 and let
ab
Q = [c d]
with ad
bc.
Then,by Lemma 2.3,
tr (Q BQ) =
+
+ c"
+ d(T1 + 2 (ac + b-c-1)
We consider two cases.
Case 1.
y = 1. Here
Re {tr(Q BQ)} =aa +
=a +
+ cc + dd + 2 Re {a.C. + b-a}
+c
= (a-a: + aE +
+d
+(a
d)
+ cZ) + (15E + b +6d + da)
=la+c12 +1b+d1
2
By Lemma 2.4 (iv), la + c 12 + 1b + d12 > 0.
Thus Re {tr (Q BQ)} > 0.
+ 12 +-a- c +
32
Case 2.
0<
<1. Here
Re {tr(Q BQ)} = a
+ b1-3 + c
+ 2 Re(a-C + ba) y
+d
>Hal2 +Ib2i +Ic12 +Id12 - 2 yIa-c-+15a1
>
I
a 12
+ ibi12 + 1cl12 +Id!12 - 2 y
(211allcl + 21b1 VI)
lal2 +1b12 + 1c12 +1d12
>+1b12 1c12
11)71)
(la
Id 12
Id 12)
= (1 - 10(1a 12 + 1b12
Since 1 -
rc12
I
> 0 and by Lemma 2. 4 (i),
ibi2
lc,
1
2
Id
12
>
it follows that
Re {tr(Q BQ)} > 0.
Hence if A is
_4426
with B, then there exist nonsingular
matrices C and D such that C1
AC = D BD. Thus
Re {trA} = Re {tr (C-1AC)} = Re {tr (D*BD)} > 0.
Lemma 2. 8.
If
x2
2
cosa
with (sin a) (cos a)
0
2
.2a
sin
1.02
33
and if z = x + iy, then
sin2 a =
Proof:
{1 - I z2I + 1z2 - 111.
Suppose
with (sin a) (cos a)
(sin
x2
2
cos2a
2
0.
a) a)
sin a
-
1
Then
x2 - (cos2a) y2
=
sin2 a cos2a.
Thus
(1-cos 2 a)x2 - (cos2a) y2 = (1 - cos a)
a)
2
cosa
So
cos 4a - (1 + x2
+
y2) cos2a + x = 0 .
Hence
2
cos a -
(1 + x2
+
y2) ± 1(1 + x2
+
y2)2 - 4 x 2
2
< 1.
We claim that
cos2a-
1+x2+y2 il+x 2 +y 2) 2 -4x2
2
For if
2
22
2
2 + y 2 +J(1 + x + y ) - 4 x
1
+
x
cos a2
2
5_ 1,
.
34
then
J (1 + x2 + y2)2 - 4x 2
< 1 - x2
- y2.
Thus we would have
x
2
+y
2
< 1 and (1 + x2
+
y2)2 - 4 x2 < (1 - x2 - y2)2
.
So
1 + 2 (x2
y2)
+
+
(x2
+
y2)2 - 4 x2 <
1-2(x2
Thus y2 < 0 which forces y = 0 and x2 < 1.
cos2a -
1+
x2 +/(1
x2)2 - 4 x 2
+
2
+
y2) + (x2 +y2)2.
So
-1
Thus we would have sin2a = 0, contradicting our assumption that
(sin a) (cos a)
Therefore
0.
sinn a = 1 - cos 2 a
=
=
If z
1
1
-
(1 + x2
+
y2) -/(1
+
x2 +y2)2 - 4 x2
2
{1 - (x2 ±y 2) + j[(1 + x)2 +y 2] [ (1 - x)2 +y 2]}
x + iy, then
I z2
iz 12
x2
2
.
35
Since
1 + z = (1 + x) + iy
12
+z
and
= (1 + x)2 +y
2
1
- z = (1 - x) - iy,
and
11
- z 12 = (1 - x)2 +y2
Thus
J
4
y1
2 [0
x)2
y1
2
)02
/I
+z
jiz2
12
11
z
12
112
z
12
Therefore
sin2a
{1-
=
1-2
z 2 + I z2
11}
Let a c C\{0}. Then
Lemma 2.9.
Re la + a-11 >0 if and only if Re {a} > 0.
Proof: Let a cC\{0}.
("Only if").
Re {a +
- 1)
Since 1 +
Suppose Re {a+ a- 1 } > 0. Then
= Re la + lai2
1>
0,
}
- Re {a +
2
} = (1 +
lal
1T) Re {a} >0.
lal
it follows that Re {a} > 0.
lal2
("lf"). Suppose Re {a} > 0. Then
Re {a-1} = Re {
2 }=
lal
12
lal
Re {a} -
1
lal
2 Re {a} >0.
36
Thus Re {a+ a-1} = Re {a} + Re {a-1}
Let A be the 2 x 2 identity matrix and B be a
Theorem 2. 7.
2 x 2 complex matrix. Then B is
B
>0
C014-
to A if and only if
p.
Proof: Let A be the 2 x 2 identity matrix and B be a 2 x
2
complex matrix.
Let B be
(1Only if").
C11/24,
2 x 2 matrices C and D such that
to A. Then there exist nonsingular
D* BD = C-1 AC= C-1 IC=
by Theorem 1. 4, B is positive definite hermitian,
("If").
Let B
1. e.
,
I. Thus,
B
(/). Then,by Theorem 1. 4, there exists a nonsingular
matrix D such that B = D* D. Thus (D-1)*B (D-1) = I = A and so
B is conjunctive with A.
Therefore,by Lemma 1. 5, B is
Oka,
to A.
Corollary 2. 7. 1. Let A be a 2 x 2 nonzero scalar complex
matrix and B be a 2 x 2 complex matrix. Then the following three
statements are equivalent.
B is
CA/J,Z,
to Ai
*
AB
E
B is conjunctive with A.
Proof: Let A be a 2 x 2 nonzero scalar complex matrix, i. e. ,
A = aI for a
0, and let B be a 2 x 2 complex matrix.
37
)4 (ii).
Suppose B is
CA/4
to A. Then,by Lemma 1. 2,
elltd, to ;1A = I. Thus,by Theorem 2. 7,
B
a1
2r1 (Th
a-1B P. Hence A B=aB= lal2 (a-1B) ra I=y_/.
.
(iii). Suppose
A*13
Then,by Theorem 1. 4, there exists
a nonsingular matrix D such that D (A B)D = I. Thus
(a D) B (a D)
(i).
B is
aI = A. So B is conjunctive with A.
Suppose B is conjunctive with A.
Chili
Then, by Lemma 1. 5,
to A.
Let B be a 2 x 2 positive definite hermitian
Theorem 2. 8.
matrix and A be a 2 x 2 complex matrix. Then A is
B if and only if A
..1frte
with
E9.
.
Proof: Let B be a 2 x 2 positive definite hermitian matrix
and A be a 2 x 2 complex matrix.
("Only if"). Let A be
..1411-6
with B. Then there exist non-
singular matrices C and D such that C-1AC = D BD. Since B is
positive definite, by Theorem 1. 4, there exists a nonsingular matrix
E such that B = EE. Thus
38
A = C (D E E D) C1
= C (C C
=
)
(D E E D) C1
(CC*) NEDC-1)* (EDC-1)]
= (CC)
Q).
Since C and Q
EDC-1 are nonsingular, CC and Q Q are positive
definite. Hence
A92.
("If"). Suppose A
6m2.
Then, by Theorem 1. 1, A is similar to a
positive definite hermitian matrix and hence to B. Thus by Lemma 1. 5
A is
__,1/ne
with B.
Corollary 2. 8. 1. Let B be a 2 x 2 unidefinite nonsingular
complex matrix and A be a 2 x 2 complex matrix. Then A is
with B if and only if (trB ) A
Let B be a 2 x 2 unidefinite nonsingular complex
Proof:
matrix, i. e. , B = EP for some P
ET= 1.
P2.
Then E
Pand some complex with
sgn (trB) and hence
*
hence (trB ) B, is positive definite.
= sgn (trB*). Thus &3, and
Let A be a 2 x 2 complex
matrix. Then A is
4kZ, with B if an only if (by Lemma 1. 2)
(t.rB ) A is
with (trB ) B if and only if (by Theorem 2. 8)
.,1416
(tr13*) A Ep2.
39
Corollary 2. 8. 2.
Let B be a 2 x 2 unidefinite nonsingular
complex matrix and A be a 2 x 2 nonscalar complex matrix. Then
A is
..,44.i.e.
with B if and only if (i) sgn (det A) = sgn (det B),
(ii) (trA) (trB*) > 0 and (iii)
(trA)2
det A
Proof:
>4.
Let B be a 2 x 2 unidefinite nonsingular complex
matrix. Then B is conjunctive with
ip
e
0
10
11
for some real p. Thus e-iPB is conjunctive with
and hence is positive definite. Suppose A is a 2 x 2 nonscalar
complex matrix. Then e
("Only if").
Let A be
PA
is also nonscalar.
with B. Since det B
0, by Lemma
1.8, sgn (det A) = sgn (det B) and (i) holds. By Lemma 1. 2, e'iPA
is _Id< with e-iPB. Thus by Theorem 2.8
e
PA
e-iPA EP2. Since
is nonscalar, tr(e-IPA) >0 and [tr(e-iPA)] 2 >4 det (jiPA) >0.
Since e iP = sgn (trB) and
= sgn (trB*),
0
<tr(e-iPA) = e- iPtrA
(sgn trB ) (trA). Thus (trB ) (trA) > 0 and (ii) holds. Also
[tr(e-iPA)]2
=
e-2113(trA)2 and det (e-IPA)
=
e-2iPdet
A.
Thus
40
e-2i13(trA)2 > 4 e -2ip det A > o. So
(trA)2> 4
det A
and (iii) holds.
("If").
Suppose A and B satisfy the conditions
(i) sgri (det B)
sgn (det A), (ii) (trA) (trB*) > 0 and (iii)
2
det A
>4.
Since e-iPB is positive definite, det (e-1113) > 0 and from (i)
det (e PA) > 0. Also from (ii) we have
0 <(trA) (sgn tr
B*) = (trA) (e-i13)
= tr (e-iPA).
From (iii) we have
e-2ip (trA)2
det (e-ipA)e-2ip det A
r tr (e-ipA)12
thus [tr (e-iPA)12 > 4 det
Theorem 2. 8, e-iPA is
(e-iPA) > 0.
..,,Ifteze,
(trA)2
det A
4
Hence e-iPA
7.--D 2.
Solby
with e-iPB and by Lemma 1. 2, A is
441,e, with B.
Theorem 2. 9.
Let B be a 2 x 2 indefinite hermitian matrix
and A be a 2 x 2 complex matrix. Then A is
only if det A <0 and (trA)2 > 0.
with B if and
41
Proof: Let B be a 2 x 2 indefinite hermitian matrix and A
be a 2 x 2 complex matrix.
with B. Then there exist 2 x 2
*
-1
nonsingular matrices C and D such that C
AC = DBD.
Since B
("Only if").
Let A be
is hermitian, D BD, and hence C-IAC, is also hermitian. Thus
trA = tr (C-IAC) is real and so (trA)2 > 0. Since B is 2 x 2
indefinite hermitian, the eigenvalues of B are of opposite signs.
Thus det B <0 and by Lemma 1. 8, det A <0.
("IVO.
Suppose det A <0 and (trA)2 > 0. Then the eigenvalues of A
are real and of opposite signs. Thus A is similar to the matrix
with a and b real and ab <0. Since
[a0
b01
is indefinite hermitian, it is conjunctive with B. Thus A is
with B.
Corollary 2.9. 1. Let B be a 2 x 2 bidefinite complex matrix
and A be a 2 x 2 complex matrix. Then A is
only if sgn (det A) = sgn (det B) and (det
B*)
.A5L-Ate
(trA)
with B if and
< 0.
Proof: Let B be a 2 x 2 bidefinite complex matrix.
Then B
is conjunctive (by Case 1 at the beginning of this Chapter) with
42
i
i(3
e
0
[0 -i1
for some real p. Thus ie -ip B is conjunctive with
-1
[0
0
11
Hence ie-ir3B is indefinite hermitian and det (ie-i13B) <0. Also
sgn (det B) =
e2,
so sgn (det B*)
= e. Let A be a 2 x 2
complex matrix.
("Only if").
Let A be ,..,416 with B. Since det B
0, by Lemma
1.8, sgn (det A) = sgn (det B). By Lemma 1. 2, ie-iPA is ..Witte,
with ie-iPB. Since ie-iPB is indefinite hermitian, by Theorem 2. 9,
[tr (ie-iPA)]2 > 0. Thus
0 < [tr (ie-iPA)]2
So e
(ie-iP)2 (trA)2 =(trA)2
-2ip (trA) 2 < 0 and hence (det B ) (trA)2
< 0.
("If"). Suppose sgn (det A) = sgn (det B) and (det B ) (trA)2 < 0.
Since sgn (det B*)
=
e-21.13, e-2iP (trA)2 < 0.
Thus 0 <
[tr (ie-i1A)]2. Since det (ie-iPB) <0, det (ie
by Theroem 2.9, ie-iPA is
(trA)2
PA)
<0. Sol
with ie-iPB. By Lemma 1.2, A
is All< with B.
Theorem 2.10. Let
B=[
1
02y 1]
43
with y > 1 and A be a 2 x 2 nonscalar complex matrix. Then A is
_A/lye with B if and only if det A > 0.
Proof: Let
0,
2y 1'
1
B=[
with y > 1 and A be a 2 x 2 nonscalar complex matrix.
("Only if"). Suppose A is }1/146 with B. Since det B = 1, by
Lemma 1. 8, det A > 0.
("If"). Suppose det A > 0. For convenience we denote 5 = + \Met A
and consider the following two cases.
Case (i): (trA)2 = 4 det A. Here the eigenvalues of A are equal.
Since A is nonscalar, A is similar to
01
r1
2-y
1-1
or to
0,
1
-5[2y
(for the same y as above).
1
Now
0
1.1
with (5-kI)
8[2y 1] is conjunctive
1
and -5[2y
0
11
(6L
4-
2y
1
01)16-kI)
r 1
0
1
1
L 2y
1
-I
0
I is conjunctive with (821) -5[ 2y 1]
1
(5
-1/2
I) =-[
1
0,
2y
11
44
which is again conjunctive (by Lemma 2.5) with
,1
0,
L 2.y
IJ
Hence A is Alivie with B.
4 det A. Here the eigenvalues of A are distinct.
Case (ii): (trA)2
We denote one eigenvalue of A by 5 a; then the other is 6a-1, and
Thus A is similar to the matrix
a-1.
a
_1
5 Fa
L2p
a
_I
with arbitrary p. Let a= tea with a real and t > 0. Then
0
-1 =6
[2p aa
5
2
Lt
0
2
0
1-
6
Lteia
2p
rte
t2
2p
is conjunctive with
t -l-ia
e
6-1/2
t-le-ial
01
t
_0
2
0
t2]
Leia
0
2p
Let p = iy 2 - sin2a. Then by Lemma 1, 10 there is a conjunctivity
having the effect
[eia
24 2
Hence A is
--d/116
sin2a
with B.
e-ia
45
Corollary 2.10.1. Let B be a 2 x 2 nonsingular contradefinite
matrix and A be a 2 x 2 nonscalar complex matrix. Then A is
with B if and only if sgn (det A) = sgn (det B).
Proof: Let B be a 2 x 2 nonsingular contradefinite matrix and
A be a 2 x 2 nonscalar complex matrix. Then B is conjunctive with
e
ip
[
1
o,
2y
1
with y > 1 and p real. The proof follows routinely from Theorem
2.10 by showingthat eA is .)1/46 ,with e -ipB if and only if
.
det (e -'1A) > 0. So by Lemma 1. 2 A is }446 with B if and only
if sgn (det A) = sgn (det B).
Theorem 2. 11.
Let
B
[
1
0
2y
1
,
with 0 < y < 1 and let A be a 2 x 2 nonscalar complex matrix. Then
A is ...,41,e, with B if and only if (i) det A > 0, (ii) Re {trA} > 0 and (iii)
2y
Proof:
2
>1+
(trA)2 - 1
4 det A
Let
B=[
with 0 <
(trA)2
4 det A
1
0
2y 1]
< 1 and let A be a 2 x 2 nonscalar complex matrix.
46
("Only if"). Suppose A is }44.4e, with B. Then there exist non-
singular matrices C and D such that C-1AC
=
D*BD.
Since det B = 1,
by Lemma 1.8, det A > 0 and (i) holds. Since
0
r 1
lj
2y
B
with 0 <y < 1, it follows from Lemma 2.7 that Re {trA} >0 and
(ii) holds. Finally, to show (iii), by Schur's Triangularization
Theorem, there exists a unitary matrix U such that
*
*
U-1 (C-1AC) U = U (D BD) U = T
is a lower triangular matrix whose diagonal entries are the
eigenvalues of A. Thus we denote the matrix T as
ia
5
2q
0
t- 1 e-ial
with t > 0, 5 > 0, a real. Further, we may assume q > 0 here.
Since 1)(T)
{62 e2ia
= 1/2
=
+
62 e-Zia - 4 62q21
62 {cos 2a - 2q2}
=6
2
{1
- 2 sin2a - 2 q 2 }
and
det T = o2,
1
2Y2
-
(3)
Idet 131
- 1 - 2 sina - 2 q2.
47
Thus y 2 = sina + q 2 > sin2a. Also, from (ii), we have
.
5
Re{teia
= Re {trT} = Re {trA} > 0.
+
Thus 5(t + t-1) cos a > 0 and hence cos a > 0.
Let
z = x + iy = k (teia +t -1 e-ia ) - trA
- ) cos a and y =
(t + t1
Then x
then x =
If sin a
(t + (1) and y = 0.
cosa
sin a.
t')
If sin a = 0,
Thus
1+
(trA)2 4 det A
=
1+
(t
=
1+ (t - t-1)2
+t1)2
(trA)2
4 det A
1
(t + t-1)2
-1
4
4
(t + t-1)2
4
4
- 0 < 2 y2
0, then
2
x2
2
1/2
(t -
1/2
.2a
sin
(t + t-1)2
2
cosa
.2a
(t - t -12
)
sin
2
4 cosa
4 sin 2a
Thus by Lemma 2.8
.2a =2 {1 +
sin
z2 -
(trA)2 -1
4 detA
z2
(trA)2
4 det A
=1
.
48
Since sin a < y 2
2y
2
(trA)2 4 det A
>1+
(trA)2
4 det A
1
which is the desired result.
("If"). Suppose A satisfies the following conditions:
(i)
det A > 0, (ii) Re {trA} > 0 and
(iii)
2
y2 > 1 +
(trA)2
4 det A
(trA)2 -1
4 det A
For convenience, we denote trA by c and \idet A by 6, and consider
the following two cases.
Case 1. c2 = 4 62. Here the eigenvalues of A are equal.
Since
det A = 62 > 0 and A is nonscalar, A is similar to
1
0
± 6 [2y 1] with y > 0. Since Re {trA} > 0 and 6 > 0, ± 5
is in fact, + 6
(61) 6 [2y
1
-1
.
A1 = 6
0
z
1 ]
1
Thus 6[ 2y
0] (6-1/21)
1
Case 2. c2
ba
1
4
62.
=
[1
2y 0]
0
1.]
01
-.1
2q a
2.y
i s conjunctive with
So A is Akvie, with B.
Here A has distinct eigenvalues, say 6aand
Then A is similar to
[a
[1
with arbitrary q > 0. Since
-
Re {trA} > 0, Re {a+ a1 } =
1
Re {trAi} =
1
5
Re {trA} > 0.
0]
1
49
Thus by Lemma 2.9 Re {a} > 0. Here we consider two subcases.
Subcase 2a. Im {a} = 0. Here a> 0 and a
cr-1
and let q = y in Al.
Then Al is conjuctive with
--
-1-2
,5
a2
a01
0
[a0 k
2
L0
2-y
1
with B.
So A is
Subcase 2b. Im {a} A 0. Let a= teia with t > 0 and a real.
x+
z
= 1/2(a+ ;1) -
Let
trA
25
Since Re {a} > 0, it follows that cos a > 0.
Here sin a = t1Im {a}
0.
Also x = 1/2 (t + t-1) cos a and y = k (t - t-1) sin a. Thus
x
2
2
.2a
sin
2
cosa
- (t
t-1)2 cos2a
2
4 cosa
(t
t-1)2 in
s2
2
4 sina
So by Lemma 2.8
sin2a
=
{1 + lz
2
(trA)2 -1
4 det A
=k
+
< 1/2
{ 2 N2}
=
-ii _
(trA)2
4 det A
-1
.
50
sin a
and hence
L[t-1/2
8
-t
=
Now' Al is conjunctive with
< Y.
[ eia
0
3-
t2
0
with q
I
Al
5
2[
2q
01
-ia
e
42 - sin2a > 0, and by Lemma 1.10,
[eia
0
2 4/y 2
[2
- sin2a e-ia
Therefore A is
104
Corollary 2.11.1.
1
is conjunctive with
.y
10
with B.
Let B be a 2 x 2 nonsingular codefinite
contrahermitian matrix and A be a 2 x 2 nonscalar complex matrix.
Then A is ..x/Ait with B if and only if (i) sgn (det A) = sgn (det B),
(ii)
(iii)
trB
Re { (trA)
IdetBIdet B
(B)
Idet BI
Proof:
tr B
5-
(trA)2
4 det A
}
(trA)2
4 det A
> 0 , and
-1
Let B be a 2 x 2 nonsingular codefinite contrahermitian
matrix and A be a 2 x 2 nonscalar complex matrix. Then B is
conjunctive (by Case 2 at the beginning of this Chapter) with
51
with 0 <-y < 1 and p real, so e
.
B is conjunctive with
r1
12y
01
1J
Also e-if3A is nonscalar.
("Only if"). Suppose A is JAZ with B.
Since det B
Lemma 1. 8, sgn (det A) = sgn (det B) and (i) holds.
e
A ts
0, by
By Lemma 1. 2,
By Theorem 2.11,
with e-iPB.
Re {tr(e-iPB)} > 0 and Re {tr(e-iPA)} > 0.
Now
trB
sgn (e -ip tr B) = sgn
eip
("pr. v. " means
2
= sgn (pr. v.
=
So
e-*= sgn
e-41
det B )
"principal value".
(trB)2 )
det B
sgn( (trB)2
det B
1
sgn (trB)
= sgn
1
tr B
= sgnIdet---r
(trBB1
sgn (
(
(trB)2det
B
(trB) (trB
Idet B I
trB
det B
(trB)2
det B
(tr B) 2
det B
)
)
52
Since Re {e-iP trA} = Re {tr(e-i(3 A)} > 0 and
multiple of
trB(
trB
\
det B
ldet
Re ((trA)
e-i13
is a positive
trB
trB
idet BI
det B
>0
and (ii) holds. Also by Theorem 2.11, we have
-)
r tr (eif? A)12
4 det (e
-i(3
4 det (e PA)
A)
(trA)2 - 1
4 det A
=
(e-43A)12
(trA)2
4 det A
i. e.
N2 <
(trA)2
4 det
.T.(B)_
2 .y2
1-2
(trA)2 - 1
4 det A
Hence
(trA)2
4 det A
(trA)2
4detA
Net B1
-1
which is the desired result (iii).
("If"). Suppose A and B satisfy the following conditions:
sgn (det A) = sgn (det B),
Re {(trA)
t.(B)
Idet BI
(
trB
trB
idet BI
det B
(trA)2
4 det A
)
} > 0, and
(trA)2
4 det A
-1
53
Since eB is conjunctive with
[12y
with 0 <y < 1, det (e
> 0.
13B)
01
Thus from (i) we have
det (e-ii3A) > 0. From (ii) we see that
Re {tr(e-43A)} = Re {e-ii3trA}, which is a positive multiple of
Re {(trA)
(
trB*
pet BI
From (iii) and the fact that
So
I - 2 y2
<
22
>
dce t( BB)
+
(trA)2 4 det A
- 1 - 2 y2, we have
(trA)2
4 det A
1
+
2
e-2ip (trA) -1
>1+
rtr (e-ipA)12 -1
=
1
.
I
(trA)2
4 det A
1
trB
det B } > 0
+
4 edet A
(trA)2
4 det A
- 2 ip (trA)2
4 e- 2ip det A
Hence we have
(3)
2y2
4 det (e 'A)
Therefore by Theorem 2.11 e-iI3A is
Lemma 1. 2, A is
with B.
rtr (e-iPA)12
4 det (e-iPA)
with e-iPB.
By
54
Theorem 2.12. Let A be a 2 x 2 nonsingular complex matrix
with distinct eigenvalues a and [3, and B be a 2 x 2 complex matrix.
Then B is 0444 to A if and only if B is conjunctive with
for some c > 0.
Proof:
This is a special case of Theorem 4. 4, proved in
Chapter IV.
Corollary 2. 12. 1.
Let
A=[
i
_i]
00
and B be a 2 x 2 complex matrix. Then the following three state-
ments are equivalent.
B is
C44,6 to A;
B is conjunctive with [ic
-101
for some c > 0;
det B > 0 and B is either bidefinite or contradefinite.
Proof: Let
i
0
A = [0 _i]
and B be 2 x 2 complex.
55
(ii):
Replace aby i and p by -i in Theorem 2.12.
Suppose B is conjunctive with [ic -1o] for some
c > 0. Then det B > 0. If c = 0, then B is bidefinite.
(iii):
If c > 0, then by Lemma 1.14 B is contradefinite.
(i):
Suppose det B > 0 and B is either bidefinite or
contradefinite. Also we have det A = 1 and trA = 0.
If B is bidefinite, then by Corollary 2.9.1 B is &Ate to A.
If B is contradefinite, then by Corollary 2.10.1 B is
C/1144
to A.
Theorem 2.13. Let A be a 2 x 2 nonscalar matrix with
det A = 1 and trA = 2, and let B be a 2 x 2 complex matrix. Then
B is
ern.' to A if and only if (i) det B
0,
(ii) det B>
(B), and
(iii) either (D (B) <- det B or Re {trB} > 0.
Proof:
Let A be a 2 x 2 nonscalar matrix with det A = 1
and trA = 2, and let B be a 2 x 2 complex matrix.
("Only if"). Suppose B is end to A. Then there exist nonsingular
*
matrices C and D such that DBD
= C-1AC. Since det A = 1,
by Lemma 1. 9, det B > 0 and (i) holds. Since A is nonscalar with
both eigenvalues 1 and by Schurts Theorem, there is a unitary
*
-1
matrix U such that U (D BD) U = U-1 (CAC)
U=
56
for some y A 0 and we may assume y > 0. Thus from the intro-
10) - 1 &action of this chapter, we see that det
B
y2 <1. Hence
(B) <det B and (ii) holds. To show (iii), we consider two cases.
2
1(2
'3-)B <- 1
y >1. Here we have 1 - 2 y <- 1. Thus det
and hence cD(B) <- det B.
Case 1.
Case 2. 0 <y < 1. Here we have by Lemma 2. 7, Re {tr B} > 0.
("If"). Suppose B satisfies the conditions: (i) det B > 0, (ii)
and (iii) either 1. (B) <- det B or Re {tr B} > 0. Then
det B >
B is conjunctive with
eiprl
2'
01
L
with y >0 or else Bis conjunctive with
e
iB, i
I.
Since det B > 0,2iI3
e= land e iB =
'
with ± [o
o
,.]
0,
0
for we would have
1.
Now B cannot be conjunctive
(B) =-det B and Re {trB} = 0,
contradicting (iii). We claim that y > 0. For if y = 0, we would
have 1)(B) = det B, contradicting (ii). Hence B is conjunctive with
± [1
2y
0]
1
with y > O.
57
2
(B)
0 < y < 1. Here det B - 1 - 2 y > - 1, and det B >0
1 and by Case 2 at the
by (i), so Re {trB} > 0 by (ii). Since e
Case 1:
beginning of this chapter satisfies Re
tr B} > 0, we must have
{e
= 1 here, and hence B is conjunctive with
rl
0,
2y
Case 2:
1J
y > 1. Here
0
[12y
11
is conjunctive (by Lemma 2. 5) with
rl.
L 2y
0,
1J
Thus in either case B is conjunctive with
rl
12y
0,
1J
which is a nonscalar matrix of trace = 2 and det = 1. Since A is
nonscalar with single eigenvalue 1, the matrix
rl
L
0,
2-y
with y > 0 is similar to A. Thus B is
elkft to A.
Corollary 2.13.1. Let A be a 2 x 2 nonsingular nondiagonable
complex matrix and B be a 2 x 2 complex matrix. Then B is eAti
58
to A if and only if
(i)
sgn (det B) = sgn (det A),
det B
(iii)
> (1)(B), and
either IT. (B) < -
det B1 or Re {(tr A*) (trB)} > 0.
Proof: Let A be a 2 x 2 nonsingular nondiagonable complex
matrix. For convenience let 8 = ildet Aland 6= sgn (trA). Then
both eigenvalues of A are 68. Thus by Schurls Theorem there is a
unitary matrix U such that
U-1AU = 88 [1
2y
for some y
0,
11
0 and we may assume y > 0. Let B be a 2 x 2 complex
matrix.
("Only if"). Suppose B is eikkt to A. Since det A
1. 9,
6
0, by Lemma
sgn(det B) = sgn (det A) and (i) holds. By Lemma 1. 2,
B is
Cif4.1 to 8-4,A. However, 8A is unitarily similar to
[
1
0
2y
11
-1_
and so 8 8 B is ei1444 to
r1
L 2,y
Thus by Theorem 2.13 we have
0,
11.
59
8-2 Idet BI
B = Idet (6
5-2 det
=
E B)
= det (6 -1_E B)
-_6 B) = 62
1
>
B) = 6-2sT.(B),
i.e., Idet B > c(B) and (ii) holds; also we have either
62(1.(B)
=
(6
E B) <- det (5 6 B) = - det (6 -1E B)
-1_
-8-2 det B I
i.e.,
(D(B) <-
or
0 <Re {tr (5-1F B)} = Re
det B I
,
=5
-1
6
16-1Z-
,
tr B}
Re {(sgn tr A*) (trB)}
-1
ItrAl
i. e.
,
Re {(trA*) (trB)},
Re {(trA ) (trB)} > 0 and (iii) holds.
("If"). Suppose A and B satisfy the following conditions:
(i)
sgn (det B)
(i1)
det B I >
sgn (det A),
(B), and
either <1.(B) <- [(Diet B j or Re {(trA ) (trB)} > 0.
Since 6 - 1_
E A is unitarily similar to
--
with y > 0, det (5 1E A) = 1. Thus by (i) we have (1) det (5 -1_E B) > 0.
From (ii) we have (2) det (5 6 B) = Idet (5& B)1= 5 -2 bet B I
>
5-2 .T.(B) = 'I. (5 -1_
E B).
60
From (iii) we have
(3) either 1)(5-1-g B) =
or Re {tr(5-1-E B)}
5-2(B)
bet 131 = - det (5-1E B)
<-
tr B}
= Re {5-1
= Re {5-1 (sgn tr A*) (trB)} > 0.
Thus by Theorem 2.13
5
-1
_
E B is
which is unitarily similar to 5
- 51
to
11
0,
1-2,/
11
6 A. Hence 5 -1 6 B is
e A and so by Lemma 1. 2, B is
to
0,14,6 to A.
We summarize the main results of this Chapter in Tables 1
and 2.
Table 1.
2 x 2 singular complex matrices.
A
B
A is 41,withB
Reference
zero matrix
complex
4=4
B=0
Theorem 2.1
complex
zero matrix
<==>
A=0
Theorem 2. 2
complex
codefinite
of rank one
<=>
det A = 0 and
sgn (trA) = sgn (trB)
Corollary 2. 3. 1
complex
contrahermitian
<=)
rank A = 1
Theorem 2. 4
of rank one
rank = 1
complex
det B = 0 and
trace = 0
rank = 1
trace i 0
complex
<
(B) < 0
det B = 0 and
either T (B) < 0
or sgn (trA) = sgn (trB)
Theorem 2. 5
Theorem 2. 6
Table 2.
2 x 2 nonsingular complex matrices.
A is
A
Corollar:- 2. 7. 1
nonzero scalar
complex
complex
unidefinite
(trB*) A
nonscalar
unidefinite
(i)
.(=> A*BE
bide finite
nonscalar
contradefinite
nonscalar
contrahermitian
<=)
E
Corollary 2. R.1
Corollary 2. 8. 2
r(i) sgn (det A) = sgn (det B)
Corollary 2. 9.1
A
2
(det B*) (trA) < 0
Corollary 2.10.1
sgn (det A) = sgn (det B)
{ (i)
sgn (det A) = sgn (det B)
trB
(ii) Re {(trA)(Idet
BI
code finite
(m) Idet(... B)
complex
p2
sgn (det A) = sgn (det B)
(ii) (trA) (trB ) > 0
(trA)2
>4
(i")det
complex
Reference
with B
(=>
BreP[ci:
(trA)2
4 det A
+
Corollary 2. 11. 1
tr B
det BP>
(trA)2
4 det A-1
for some p > 0
Theorem 2.12
Table 2. Continued.
A
B
[io
nondiagonable
A is AC., with B
complex
det B > 0 and B is either
bidefinite or contradefinite
complex
(i)
<=>
sgn (det B) = sgn (det A)
B I> t.(B)
either I, (B) < - Idet B
or Re {(trA*) (trB)} > 0
Symbols:
f If-go!
denotes "is similar to"; 4V-11 denotes "is conjunctive with".
Reference
Corollary 2. 12. 1
CorollarN,- 2.13.1
64
n x n COMPLEX MATRICES
WITH REAL EIGENVALUES
In this chapter we consider only certain classes of n x n
complex matrices whose eigenvalues are real. These matrices
which will be discussed are: hermitian matrices (since hermitianness is invariant under conjunctivity), diagonal matrices (or
diagonable matrices, since diagonability is invariant under
similarity), and triangular nondiagonal matrices. Although a few
specific results have been proved in Chapter II for 2 x 2 matrices,
we restate these results here for completeness.
§ 3.1.
Diagonable Matrices
We restate Theorems 2. 7 and 2. 8 for the n x n complex
matrices
Theorem 3.1. Let A be the n x n identity matrix and B be an
n x n complex matrix. Then B is a/n/L to A if and only if B
Proof: See Theorem 2. 7.
Theorem 3. 2.
Let B be an n x n positive definite hermitian
matrix and A be an n x n complex matrix. Then A is ..,1-#6 with B
if and only if A E p2.
65
Proof: See Theorem 2. 8.
Theorem 3. 3. Let B be an n x n hermitian matrix and A be an
n x n complex matrix. Then A is .)Me,, with B if and only if A is
similar to a hermitian matrix of same rank and signature as B.
Proof: Trivial.
Theorem 3. 4. Let B be an n x n hermitian matrix and A be
an n x n normal matrix. Then A is
_ill/6
with B if and only if
A is conjunctive with B.
Proof: Let B be an n x n hermitian matrix and A be an n x n
normal matrix.
("Only if"). Suppose A is
_Ike. with B.
Then there exist non-
singular matrices C and D such that C-1 AC = D* BD. Since B is
hermitian, D*BD,an.d hence C-1AC, is also hermitian and is there-
fore normal. Hence A and C-1AC are (by Theorem 1.6) unitarily
similar. So A is conjunctive with D BD, hence with B.
("If"). Suppose A is conjunctive with B. Then by Lemma 1. 5
A
is ..V./kLZ.,with B.
Theorem 3. 5.
Let A be an n x n real diagonal matrix with
positive distinct eigenvalues and B be an n x n complex matrix.
Then B is ejn4, to A if and only if B EP 3.
66
Let A be an n x n real diagonal matrix with positive
Proof:
distinct eigenvalues and B be an n x n complex matrix.
("Only if").
ei,d, to A. Then there exist nonsingular
Suppose B is
matrices C and D such that
D* BD = C-1AC.
Thus by Schur's
Triangularization Theorem there exists a unitary matrix U such that
*
*
U (D BD) U = U-1(C-1AC)U = T is a lower triangular matrix whose
diagonal entries are the eigenvalues of A. Since A has positive
eigenvalues, B is conjunctive with a lower triangular matrix of
positive diagonal. Therefore, by Theorem 1. 2, B E 9)3.
("If"). Suppose B 9-)3.
Then, by Theorem 1. 2, there exists a
nonsingular matrix P such that P*BP is a lower triangular matrix of
positive diagonal. Thus there exists a nonsingular positive diagonal
*
*
matrix Q such that Q (P BP)Q= (PQ) B (PQ) is a lower triangular
matrix whose diagonal entries are the same as those of A. Since
the diagonal entries of A are distinct, (PR) B(PQ) is similar to
A.
Hence B is
en"-
Theorem 3. 6.
to A.
Let A be an n x n nonsingular complex matrix
similar to a real diagonal matrix and B be an n x n nonsingular
cohermitian matrix. Then B is entL to A if and only if B is
hermitian and sig B
Proof:
sig A.
Let A be an n x n nonsingular complex matrix similar
to a real diagonal matrix and B be an n x n nonsingular cohermitian
67
matrix. ("Only if"). Suppose B is Cliut to A. Then there exist
nonsingular matrices C and D such that D*BD = C-IAC. Since B is
nosingular cohermitian, B = EH, where H is nonsingular hermitian
and E is a nonzero complex number. Hence there exists a unitary
* *
-1 -1
U HI is a diagonal
matrix U such that U (D 8 HD)U U(CAC)
matrix, and this diagonal matrix must be real because all the eigenvalues of A are real. Thus HI is hermitian and nonsingular, E must
be real and so B must be hermitian. Thus, by Theorem 1. 3,
index (B) = index (H1) = index (A) and hence sig B
("If").
Suppose B is hermitian and sig B
sig A.
sig A. Since there exists
a nonsingular matrix C such that C-1AC is a nonsingular real
diagonal matrix, C-1AC is nonsingular hermitian and
sig (C 'AC)= sig B. Thus, by Theorem 1. 3, B is conjunctive with
C-IAC and hence B is entd., to A.
We next prove a Lemma.
Lemma 3.1. If S is n x n complex lower triangular
nondiagonal with all diagonal entries real and two of them of
opposite signs, then S is contradefinite.
Proof:
Let S be n x n complex lower triangular nondiagonal
with all diagonal entries real and two of them of opposite signs.
First, suppose S is nonsingular. Since S is lower triangular
nondiagonal, S is contrahermitian and has all diagonal entries on
the real line through 0 and not all on one side of 0. Hence, by
Lemma 1.14, S is contradefinite. Next, suppose S is singular of
68
*
rank r and codefinite. Then, by Lemma 1.12, S is -regular. So,
by Lemma 1.13, S is unitarily conjunctive with the direct sum of a
nonsingular r x r matrix S1 and a zero matrix. Since S is lower
triangular nondiagonal, S is contrahermitian. Thus the direct sum,
and hence
S1'
is also contrahermitian. Since S has real diagonal
entries with two of them of opposite signs, the direct sum, and
hence
S1'
has real eigenvalues with two of them of opposite signs.
Thus by Schur's Theorem S1 is unitarily conjunctive with a lower
triangular nonsingular matrix S2 with real diagonal entries and
two of them of opposite signs. S2 is also contrahermitia.n, and
hence nondiagonal. Thus S2 is contradefinite (as in the first case).
Therefore S1' and hence 5, is also contradefinite, giving a
contradiction.
Theorem 3. 7.
Let A be an n x n nonsingular real diagonal
matrix with distinct eigenvalues and B be an n x n nonsingular
nonhermitian codefinite matrix. Then B is
if either A (.." and B
Proof:
or else A
E-
eind, to A if and only
rTh3
and B e-t3-
.
Let A be an n x n nonsingular real diagonal matrix
with distinct eigenvalues and B be an n x n nonsingular nonhermitian
codefinite matrix.
("Only if"). Suppose B is ell4Z to A. Then there exist nonsingular
matrices C and D such that D BD = C-1AC. Thus by Schur's
69
Theorem there exists a unitary matrix U such that
U (I) BLI\U = U-1(C-1AC)U = T is a lower triangular matrix whose
diagonal entries are the same as those of A. Since B is
nonhermitian and T is conjunctive with B, T is also nonhermitian,
and hence nondiagonal.
First we claim that A Epu
e.,
A is positive definite or -A is positive definite. For if A q-Dti-P,
then A has real diagonal entries with two of them of opposite signs,
so T also has real diagonal entries with two of them of opposite
signs. Thus, by Lemma 3.1, T is contradefinite. Since B is
conjunctive with T, B is also contradefinite, contradicting the fact
that B is codefinite. Hence we must have A
cpu _p. Therefore
either A EPand, by Theorem 3.5, B ER3, or A E _p , i. e.,
-A Ep; so, by Theorem 3. 5, -B EP3 and hence B
("If").
Suppose A EPand B
E
(P3.
to A. Suppose A E_p and
Then,by Theorem 3. 5, B is
B -y. Then - A ePand
-B eP3. Thus, by Theorem 3. 5, -B is 6/14Z to -A. Therefore,
by Lemma 1. 2, B is 0/41.4 to A.
Corollary 3. 7.1. Let A be an n x n nonsingular complex
matrix whose eigenvalues are real and distinct, and let B be an
n x n nonsingular contrahermitian codefinite matrix. Then B is
6,4 to A if and only if either A EP2 and B EP3 or else
A E - (P2 and B e -CP3.
70
Proof: Let A be an n x n nonsingular complex matrix whose
eic-,envalues are real and distinct and let B be an n x n nonsingular
contrahermitian codefinite matrix. Since A is similar to an n x n
nonsingular real diagonal matrix with distinct eigenvalues, by
to A if and only if either A is similar
Theorem 3. 7,B is
to an element of sand B
or else A is similar to an element
of _9D and B e -F3, if and only if (by Theorem 1.1) either
A eP 2 and B
or else A E
2
and B e -
Although an idempotent matrix is not necessarily diagonal,
it is similar to a diagonal matrix.
Theorem 3. 8. Let A be an n x n idempotent matrix (A = A)
of rank r, 0 < r <
n.
Let B be an n x n complex matrix. Then B
is &//42 to A if and only if B is conjunctive with
,0
21
01
0
e I r-s
EDI
On- r-
s
for some s < r.
Proof: Let A be an n x n idempotent matrix of rank r,
0 < r < n. Let B be an n x n complex matrix.
("Only if"). Suppose B is
efriz to A. Then there exist non-
singular matrices C and D such that D BD = C-1AC. Since A is
idempotent of rank r, A is similar to the matrix
71
[ Ir 0]
0
0
Thus by Schur's Theorem there exists a unitary matrix U such that
.1,
U (D BD)U = U-1(C-1AC)U
=
1T
L
01
V
where T is arxr lower triangular matrix all of whose diagonal
entries = 1 and V is an (n - r) x r matrix. Since rank T = r and
rank A = r, the lower right block W must be On-r. Since
[T
V
TO ,
LV
01
0
0
n-r
is idempotent,
rT
rT
r
LV
0-1 LV
L
T2 01
VT 0
-I
Therefore T2 = T so T = I (and V is arbitrary),
01
i.e.,1(C-1AC)U = [IrV
is conjunctive with
P
0
0
0
Now,
n-r
Ir
V
01
[Ir
V
Oi
po
0
0E
0
n- r
[Ir
01
LE *VP
0
j
for every r x r unitary matrix P and every (n - r) x (n - r) nonsingular matrix E. Let s = rank V. Then we can here choose E
nonsingular so that 1/2 E V is in row-echelon form and then choose
P as a permutation matrix so that E VP =
2Q1
o
[2Is
oi
72
where Q is some s x(r - s) matrix. Thus we may assume V itself =
[I r
2Q
[2Is
x s(r-sj
°(n-r-s)
So
0
V
0
0
9 0 n- r- s. Now,
0
21
-21
0
I
-2Q-2Q
Is
r-s
..-
,
I
s
s
0
0
0
0
0
0
0
0
0
0
-2Is
-Q
I
0
0
-2Q-'
I
0
Is
I
0
0
, which is again conjunctive
0
*
I
r- S
(1
+QQ
r-+Q
s
[021 s
Q)
0
21s
+Q Q)
0
0
0
0
0
0
0
0
*
r- S
0
r-0
0
(I
-4(I5+QQ )
0
0
0
0
s
r-]
0
)
Is
)
0
conjunctive with
0
r-s
2Q
0
-1/2(I5+QQ*)-1
Is
-12-(Is+QQ
0
r-s
s
0
with
I
02Q
-4(Is+QQ
0
0
2Q2Q
*
0
0
0
r-s
n-r
2Is
1i s
01
0
s
0
I
r -s
00
0
0
01
I
r-s +Q*Q
73
Since B is conjunctive with
[0
01
e r-s e
0
21s
r
V
0
B is conjunctive with
On-ri
n-r - S.
[0
("If"). Suppose B is conjunctive with
21s
01
r-s @0 n- r- s,
ED
0
0
with 0 < s < r < n. Since
I
0
r-S
0
215
(3
01
e I r-s is conjunctive with
0
0
Is
_2Is
Is
B is conjunctive with
0
0
0
,_2Is
s
0
0
-
0
e 0 n- r - s.
0
Thus there exists an n x n nonsingular matrix D such that
Is
D BD =
0
2Is
0
0
I
0
r-s
0
@0 n-r-s. Hence D BD is an idempotent
Os
matrix of rank r. Since A is also an n x n idempotent matrix of
rank r, D BD is similar to A. Thus B is CALL to A.
§3. 2.
Triangular Matrices
We weaken the hypotheses of Theorems 3. 5, 3. 6 and 3. 7 by
allowing the matrix A to have repeated eigenvalues. It is not clear
whether necessary and sufficient conditions can be found so that the
74
matrix B is CAZ to A. However, we have the following extension
of Theorems 3. 5, 3. 6 and 3. 7, by considering a class of matrices
similar to A. For X. ER (R denotes the real field), we define
1
minus the
number of negative X ..1 When A is similar to diag (X 1,X 2,
sig A = sig (X1, X 2,
.
n.
,
.
, X n).
Let A1
Theorem 3.9.
i = 1, 2,
.
Let
2'
aAbe the set of all n x n matrices Similar to
triangular matrices of diag =A. Let B be an n x n complex matrix.
Then there exists A E
which is ."Ime with B if and only if
B
Proof: The proof is a slight modification of the proof of
Theorem 3. 5.
Theorem 3.10. Let A (x. 1, X 2,
for i
1, 2,
.
.
.
, n.
Let
.
.
, X n)
withX.ERVO}
be the set of all n x n nonsingular
matrices similar to triangular matrices of diag = A
Let B be an
n x n nonsingular cohermitian matrix. Then there exists A e
which is
with B if and only if B is hermitian and A is
diagonable and sig B = sig A .
Proof: The proof is a slight modification of the proof of
Theorem 3. 6.
75
Theorem 3. 11.
1, 2, .
i
.
,
n.
Let A= (x
x 2,
.
.
. ,x n) with X
E
eVO for
Let a be the set of all n x n nonsingular matrices
similar to triangular matrices of diag = A
.
Let B be an n x n
nonsingular contrahermitian codefinite matrix. Then there exists
with B if and only if either sig
which is
A
B
e?3 or else sig A= -n and B c
n and
rTh3
.
Proof: The proof is a slight modification of the proof
of Theorem 3.7.
An n x n complex matrix S is called nilpotent if A
= 0 for
some positive integer m.
Theorem 3.12.
Let A be an n x n nilpotent matrix of rank
n- 1 and let B be an n x n complex matrix. Then B is e/14.Z to A
if and only if B is conjunctive with a strictly lower-triangular
matrix of rank n Proof:
1.
Let A be an n x n nilpotent matrix of rank n - 1 and
let B be an n x n complex matrix.
("Only if"). Suppose B is
6,11./Z
to A.
Then there exist nonsingular
matrices C and D such that D BD = C-1AC. Furthermore, by
Schur's Theorem there exists an n x n unitary matrix U such that
*
U (D BD)U = U-1(C-1AC)U = T is a lower triangular matrix. Since
A is nilpotent of rank n - 1 and T is similar to A, T is strictly
76
lower triangular of rank n -
1.
("If "). Suppose B is conjunctive with a strictly lower triangular
matrix T of rank n - 1. Since A is nilpotent of rank n - 1, A and T
have the same invariant factors. Thus T is similar to A and hence
B is
eAkt to A.
77
IV. n x n COMPLEX MATRICES
WITH COMPLEX EIGENVALUES
In this chapter we again consider only certain classes of n x n
complex matrices whose eigenvalues are complex numbers. These
matrices include diagonable matrices, triangular nondiagonal
matrices, unitary matrices, positive stable matrices and accretive
matrices.
§4. 1.
Diagonable Matrices
Theorem 4.1. Let A be an n x n nonzero scalar complex
matrix and B be an n x n complex matrix. Then the following three
statements are equivalent.
to A;
B is
*
c,Th
A B e7-';
B is conjunctive with A.
Proof: See Corollary 2. 7.1.
Theorem 4. 2.
Let B be an n x n unidefinite nonsingular
complex matrix and A be an n x n complex matrix. Then A is
with B if and only if (trB ) A
F2.
Proof: See Corollary 2. 8.1.
78
Let A be an n x n nonsingular complex matrix
Theorem 4. 3.
all of whose eigenvalues are distinct and B be an n x n nonsingular
contradefinite complex matrix. Then B is efrkt to A if and only
if sgn (det B) = sgn (det A).
Proof: Let A be an n x n nonsingular complex matrix all of
whose eigenvalues are distinct and B be an n x n nonsingular contradefinite complex matrix.
("Only if"). Suppose B is CM," to A. Since A and B are nonsingular, by Lemma 1. 8 sgn (det B) = sgn (det A)..
("If"). Suppose sgn (det B) = sgn (det A). We denote the eigenvalues
of A by t1, t2,
.
t and hence
tn. Thus det A = t1 t2
. ,
.
sgn (detB) = sgn (t1 t2
.
.
.
tn). Hence by Lemma 1.15 B is
conjunctive with a lower triangular matrix T such that
diag T = (t1, t2,
.
.
,
tn). Since t1, t2, .
similar to A. So B is
.
, tn are distinct, T is
to A.
Theorem 4. 4. Let A be an n x n complex matrix similar to a
diagonal matrix with exactly two eigenvalues a and p, with rank
(A - p1) = r, 0 < r < n.
Let B be an n x n complex matrix. Then B
is efr1.4 to A if and only if B is conjunctive with
S
a
0
j=1
Y
R
9
al r-s e p in- r - s
79
for some s < min {r, n-r} and for some positive y l' Y2'
Let A be an n x n complex matrix similar to a
Proof:
diagonal matrix with exactly two eigenvalues a and p, with rank
Let B be an n x n complex haatrix.
(A - pI) = r, 0 < r < n.
("Only if"). Suppose B is CM/4, to A. Then there exist non*
singular matrices C and ID such that DBD
= C-1AC. Since A is
similar to a diagonal matrix with exactly two eigenvalues aand p,
with rank (A - pI) = r, so is C-1 AC and hence
D* BD.
Thus by
Schur' s Triangularization Theorem there exists a unitary matrix U1
such that
*
U1
0
(D BD)U = [air
V
*
with rank V = a for
pin_r
1
some s < min {r, n-r}. Since V is an (n-r) x r matrix with rank s,
by Autonnels Lemma (Theorem 1. 8) there exist arxr unitary
matrix Q and an (n-r) x (n-r) unitary matrix P such that
00i
P VQ =
,
where As is an s x s diagonal matrix with each
diagonal entry Nj > 0, j =
Then
U2
1,
2,
,
s.
Q
Now let U2 = [0
is unitary. Thus
0
[otI
*
U2 U1
.
(D BD)U1U2 =
0
PO
V
pin_r
Q
0
0
P
0
pi.
80
0
P VQ Pin- r 1
Since P*VQ
,
=
o
0
o
0
(DU1U2)* B(DU1U2) =
0
S
o
pin-r- s
s_
,e,
0
0
0
0
0
I
0
0
Is
0
0
Is
let U3 =
s-
*
*
Then U3(DU1 U2) B(DU1U2)U3 =
=
r-s
0
0
0
In-r- s
aIs
As
0
0
0
pis
o
0
o
0
aI r- s
0
0
0
0
pin-r- s ..,
,
e aI r- s
1
El)
pin-r-s.
81
1
0
0
2
Since As =
,
aI
conjunctivity
by a permutation
o
conjunctive with
_As
i3Is
Therefore B is conjunctive with
(
s
\j10
[a
y.
0
)
C) aI r-s
e
pIn-r-s
r3
for some s < min {r, n-r} and some positive y1,1 v' 2'
("If").
Suppose B is conjunctive with
B1=
jc:?1
[
a
01 )
Yi
P
'1r -s
e
for some s < min {r, n-r} and some positive
Let
C=
(
J=1
(s)
Then
C-1=
( j=1
)(:) in-2s
Yr 112,
82
Thus
-1
B1C
o
Clin-2s} (c)'
j=1
e
ocar-spIn-r_
y
0
1
0
j=1
1
0
1
al r-s 0 fl In- r- s
0
( j=1[ y.
P- a
(
s [a
0
)133,
j=1
0
p
al r-s
0 13
In- r- s
,
which is similar to
air 0 p In-r , which in turn is similar to A. Hence B is e/14'L
to A.
An n x n complex matrix S is called involutory if
S2 = I.
Corollary 4. 4.1. Let A be an n x n involutory matrix with
rank (A + I) = r, 0 < r < n. Let B be an n x n complex matrix. Then
B is e/P1.4/ to A if and only if B is conjunctive with
83
r-s
(-In-r-s)
for some s < min {r, n-r} and for some positive yl, y2,
Proof: Replace a by "1" and p by "-1" in Theorem 4.4.
§4. 2.
Triangular Matrices
In this section we consider those matrices which are not
conjunctive with diagonal matrices.
Theorem 4. 5. Let A be an n x n singular complex matrix all
of whose nonzero eigenvalues are distinct and such that A has only
linear elementary divisors at 0, and let B be an n x n *-irregular
complex matrix. Then B is ei444 to A if and only if
rank B = rank A.
Let A be an n x n singular complex matrix all of
Proof:
whose nonzero eigenvalues are distinct and such that A has only
linear elementary divisors at 0, and let B be an n x n *-irregular
complex matrix.
("Only if"). Suppose B is
0/11./i
to A. Then by Lemma 1. 7
rank B = rank A.
("If").
Suppose rank B = rank A. Let r = rank A with 0 < r < n,
and t1, t2,
.
.
, tr be the nonzero eigenvalues of A. Since B is
84
*-irregular of rank r, by Lemma 1.16 B is conjunctive with a lower
with diag T = (t t2
triangular matrix [TV 0]
0
.
.
.
t
r) and for some
(n-r) x r matrix V. Since A has only linear elementary divisors at 0,
the matrix [VT
0
]
0
.
similar to A. Hence B is e/41,1 to A.
For the case where B is an n x n singular *-regular contradefinite matrix, by Lemma 1.131B is *-regular if and only if B is
conjunctive with the direct sum of a nonsingular matrix B1 and a
zero matrix, and the conjunctivity class of any such B1 is uniquely
determined by B.
Thus
n
(B) = sgn (det B1) is defined and
conjunctively invariant for every *-regular B (if we define n(0)
Thus
n
= 1).
(B) is the signum of the product of nonzero eigenvalues of B.
We state without proof the following result.
Fact 4.1. Let A be an n x n complex matrix of rank r
(0 < r < n) such that its nonzero eigenvalues X 1,X
2,
.
are
distinct and it has only linear elementary divisors at 0. Let B be
an n x n *-regular contradefinite matrix. Then B is e/kut to A if
and only if rank B = r and 7-1 ( B) = sgn (X 1X2...X r).
An n x n complex matrix A is said to be nonderogatory if and
only if the minimal polynomial of A is equal to its characteristic
polynomial. It is also known that A is nonderogatory if and only if
85
the Jordan form for A has just one block for each characteristic
root of A. We are unable to find (useful) necessary and sufficient
conditions for an n x n complex matrix B to be CMAt to an arbitrary
given nonderogatory matrix A. However, we have solved the prob-
lem for a nonderogatory matrix A with only one characteristic root
(see Theorem 4. 6). We first introduce some preliminary lemmas.
Lemma 4.1.
Let S be a 3 x 3 nondiagonal lower triangular
matrix with diag S = (1,1,1). Then S is conjunctive with a 3 x 3
lower triangular matrix T with diag T = diag S and such that the
first subdiagonal of T has no zero entries.
Proof: Let S be a 3 x 3 nondiagonal lower triangular matrix
with diag S = (1,1,1). We consider the following three cases.
Case 1.
-1
S is of the form
w
_0
0
O-
1
0
0
,
w
0.
1_
Here we apply to S the [2, 3] - subconjunctivity defined by the matrix
0
*- 1
1
-5
, 4
3
4]
and get thereby a matrix T 1=
4
5
w
3
-w
0-
1
0
0
1
We further apply to T1 the [1, 2]-subconjunctivity defined by the
86
[1
o
and get thereby a matrix T
matrix
w1
4
0
6
1
0
--5
w2
where
w1
=
4
5
0 and
w2
Case 2. S is of the form
=
5
0
w
o.
0
1
0.
g
,
Here we apply to
S the [1,2] -subconjunctivity defined by the matrix
0
3
0
1
4
g
-34] and
4
[3
0'
0
get thereby a matrix T2 =
1
g
We then apply
.
1
[4
5
to
T2
the [ 2, 3] -subconjunctivity defined by the matrix
1
and get thereby a matrix T =
g2
=
5
g
g1
0
0
1
0
g2
1
where g1 =
3
g
0 and
0.
^1
Case 3. S is of the form
0
0
0
1
0
x
0
1
,x
0.
Here we apply to S
the interchanging [2,3] -subconjunctivity and get thereby a matrix
87
'l
x
0
0
1
0
0
1
,
which is covered by Case 1.
Lemma 4. 2.
Let
000
1
00
10
1
with xy
0.
Then S is conjunctive, by a
vw 01
[ 2, 3, 41 -subconjunctivity, with a lower triangular matrix T of the
1
00
0
.x1
1
0
0
with x'y'z'
form
ul
V
y`
1
W
0.
0
1
Proof: Let
1
000
1
00
10
with xy
0.
vw 01
We consider the following three cases.
Case 1. w
0, u = 0.
Here we apply to S the [ 2, 3] -subconjunctivity defined by the matrix
88
1
ry
x'
11
and get thereby a matrix T =
LI'
000
1
0
0
[ III
yi
1
0
V1
W'
z'
with
1
._.,
x'y' z'
0, since x' = xy , y' = y and z' = w.
0, u
Case 2. w
0.
Here we apply to S the [2, 3] -subconjunctivity defined by the matrix
0
11
y
x'
00
0
0
0
1
0
z
1
1
and get thereby a matrix T =
UI
V1
0, since x1 = -
with xlyizt
,
W1 w
y = y and zr = w.
Case 3. w = 0.
Here we apply to S the [3, 4] - subconjunctivity defined by the
matrix
1
4
3
[-3
4]
and get thereby a matrix
00
00
y1
0
1
1
T=
5
*
3
5
4
0
3
3
Y
0
1
Case 1 if u= 3v
4
which is covered by Case 2ifu
4
89
Lemma 4. 3.
Let S be an n x n nondiagonal lower triangular
matrix with diag S = (1,1, ... ,1). Then S is conjunctive with a lower
triangular matrix T with diag T = (1,1, ... ,1) and such that the first
subdiagonal of T has no zero entries.
Proof: We use induction on n, and we state the induction
assertion, Pn, as a sublemma.
Sublemma 4. 3. 1.
(= induction assertion, Pn). Whenever
S is n x n lower triangular nondiagonal of diag S = (1,1, ... , 1), then
S is conjunctive with a lower triangular matrix T of
diag T = (1,1, ... , 1) such that the first subdiagonal of T contains no
zero entries.
Proof of Sublemma 4.3.1.
and
P3
P1
is vacuous' P2 is trivially true,
is true by Lemma 4.1. So we assume n> 4 and Pn-1 is true.
Let S be n x n lower triangular nondiagonal of diag S = (1,1, ... , 1).
Since
s= S[nin)
_S(n)
01
1
we consider the following two cases.
Case 1. S(n) is (n-1)x (n-1) nondiagonal.
From our induction
hypothesis, S(n) is conjunctive with a lower triangular matrix T1
whose diagonal = (1,1, ... ,1) such that the first subdiagonal of T1
90
contains no zero entries. Thus S is conjunctive with a matrix
[Ti 01
T2 =
V
1
for some 1 x (n-1) matrix V. Now T2 [n-3, n-2, n-1, n] is lower
triangular nondiagonal of diag
(1,1,1,1).
Then we apply to T2 the
[n-2, n-1, n] -subconjunctivity defined in Lemma 4. 2 and get
thereby a matrix T such that T[ n-3, n-2, n-1, n] is the matrix
1
0
0
0
xl
1
0
0
with xty'z'
1
VI
wtZ
0.
Thus we have a
0
1
triangular matrix T of diag T = (1,1, ... ,1). Except possibly for the
last three entries, the first subdiagonal of T is the same as that of
T2, so all entries on the first subdiagonal of T are nonzero.
Case 2. S(n) is diagonal.
Here we have S[nin) 1 0. We consider two subcases.
Subcase 2a. Sn,
n-1
=x
0.
Here S[n-2, n-1, n] is of the form
1 0 01
010
91
with x
We apply to S the [ n-2, n-1, n] -subconjunctivity defined
0.
in Lemma 4.1 and get thereby a matrix T3 such that
T3[ n- 2, n-1, n] =
Since
T3(n)
1
0
0
yr
1
0
x'
1
with xlyt
0.
is nondiagonal, T3 is a matrix covered by Case 1.
Subcase 2b. S[nin-1, n)
0 and
Sn, n-1
= 0.
Suppose Sn, r = x
0
for some r such that 1 < r < n -2, and furthermore such that this
is the last nonzero off-diagonal entry in the nth row, i.e. , such
that S[ n Ir+1,
,
n-11 = 0. We then apply to S the interchanging
r+1, n] -subconjunctivity and get thereby a matrix T4 such that
is lower triangular nondiagonal. Thus the matrix T4 is again
covered by Case 1.
T4(n)
Theorem 4. 6.
0 and n> 2 and let A be an n x n
Let X.
nonderogatory complex matrix with X as its only eigenvalue
(i. e. , A - X I is nilpotent of rank n-1).
Let B be an n x n complex
matrix. Then the following three statements are equivalent.
(i).
B i s eiL to A;
(ii)
B is conjunctive with a lower triangular nondiagonal matrix
with diag = (X
(iii).
B
E
,X,
,X );
xcp 3 but not Exp
.
92
0 and n> 2 and let A be an n x n nonderogatory
Proof. Let X
Let B be an n x n
complex matrix with X as its only eigenvalue.
complex matrix.
4 (ii).
Suppose B is C.A4 to A. Then there exist nonsingular
matrices C and D such that D BD = C-1 AC. Since A is non-
derogatory with X as its only eigenvalue, by Schur's Theorem
there exists a unitary matrix U such that
*
-1 -1
U (D BD)U = U(CAC)
U=T
is a lower triangular nondiagonal matrix with
diag T = (X , X ,
,X). Hence B is conjunctive with a lower
triangular nondiagonal matrix with diag = (X , X ,
(i).
, X ).
Suppose B is conjunctive with a lower triangular non-
diagonal matrix with diag = (X , X ,
,
).
Then X -1B is
conjunctive with a lower triangular nondiagonal matrix with
diag = (1,1,
... , 1). Thus,by Lemma 4.31X
1B
is conjunctive
with a lower triangular matrix whose diag = (1,1, ... , 1) and
whose first subdiagonal has no zero entries. Hence B is
conjunctive with a lower triangular matrix T with
diag T = (X ,X,.
) and the first subdiagonal of T having
no zero entries. So T is nonderogatory with only one nonzero
eigenvalue X. Since T and A have the same invariant factors,
T is similar to A. Therefore B is ein,6 to A.
93
The following statements are clearly equivalent.
(ii)<=j(iii).
B is conjunctive with a lower triangular nondiagonal matrix T
with diag T
X
(X
-1B is conjunctive with a lower triangular nondiagonal matrix
V with diag V = (1, 1,
X
, X ).
,X,
-1
BE
3
(".
B EXP
3
but X
.
.
, 1).
-1B OD.
but B xp .
(a)<=>(b).
Trivial.
(b)44(c).
By Theorem 1.2 and the fact that V is nondiagonal.
(c)<4(d). Trivial.
§4.3. Unitary Matrices
In this section we restrict ourselves to a special class of
unitary matrices. It is not clear at present that we can completely
solve the problem for unitary matrices. We first state a wellknown result.
Theorem 4. 7. (Polar Factorization Theorem,
[10], pp. 74-75; [12], p. 357) Let S be an n x n nonsingular
complex matrix. Then there exist unique n x n positive definite
hermitian matrices P and Q and n x n unitary matrices U and V
such that S = PU = VQ.
Furthermore, we have U = V.
94
An application of Polar Factorization Theorem is the following
theorem which was pointed out by M. -D. Choi. (see e. g. [5],
Theorem 7 (iii)).
Let A be an n x n nonsingular complex matrix.
Theorem 4. 8.
Then there exists a unitary matrix B such that B is both
CAZ
to, and AlAte with, A.
Proof: Let A be an n x n nonsingular complex matrix.
Then
by Theorem 4. 7 there exists a positive definite hermitian matrix P
and a unitary matrix B such that A = PB. Since P is positive definite,
by Theorem 1. 4 there exists a nonsingular matrix D such that
P = DD . Thus A = PB
DD B = D(D*BD)D 1.
and hence B is eAira to A.
So A = (C
- *
-
(C 1)B
=
Now let D*
(C-1)*C-1BCC-1.
=
C-1 .
So D BD = D lAD
-1*
Then P = (C)(C-11_).
Thus C-1BC
C AC and
so B is _14444 with A.
An n x n complex matrix S is said to be a symmetry if
S* = S
-1
= S.
Theorem 4.9. Let A be an n x n unitary matrix and B be an
n x n nonsingular hermitian matrix. Then the following three
statements are equivalent.
A is At#16 with B;
A is a symmetry and index A = index B;
(iii)
A is conjunctive with B.
Proof: Let A be an n x n unitary matrix and B be an n x n
nonsingular hermitian matrix.
(ii). Suppose A is _JAC with B.
matrices C and D such that C-1 AC
Then there exist nonsingular
D* BD.
Since B is
hermitian, D BD,and hence C-1 AC, is also hermitian. Thus
l
C AC
_1
(C
AC)
*
*
= C A (C
-1* . So A(CC ) = (CC )A
)
.
Since
is positive definite and A and A are unitary, by
CC
Theorem 4. 7 we have A = A. Together with the unitary
property of A, this gives us A-1
=
A* = A.
So A is a
*
symmetry. Also index A = index (C-1AC) = index (D BD)
= index B.
Suppose A is a symmetry and index A = index B. Since
A-1
= A = A, A is hermitian and nonsingular, So A is
conjunctive with B.
Suppose A is conjunctive with B. Then by Lemma 1. 5
A is
..,,,hfrze,
with B.
We state without proof a dual of Theorem 4.9.
Theorem 4.10. Let B be an n x n unitary matrix and A be an
n x n involutory matrix. Then the following three statements are
equivalent.
96
B is CAZ to A;
B isa symmetry andindex B = index A;
B is similar to A.
§4. 4.
Positive Stable Matrices
We denote by0, the set of all n x n complex matrices S
such that every eigenvalue of S is in ORHP (the open right-half
plane). An element of / is called a positive stable matrix. We
also denote by ,a , the set of all n x n complex matrices S such
that S + S is positive definite.
An element of
is called an
accretive matrix. A well-known result of Lyapunov may be stated
in the following form.
Theorem 4.11. (Lyapunov, [3], [4], [14]).
cz_pg=0P
A simple application of the above theorem can be stated in the
following result, Theorem 4.12.
Theorem 4.12. Let A be an n x n positive stable matrix.
Then there exists an n x n accretive matrix B such that B is both
to, and
with, A.
Proof: We use Theorem 4.11 and the proof is the same as that
of Theorem 4. 8.
97
In view of Theorems 4. 8 and 4.12, we have the following
generalization.
Theorem 4.13.
Let
a=
.
there exists
Then for Ae
to, and likte with, A.
B EB such that B is both
Proof: See the proof of Theorem 4. 8.
eft" to, and
However, it is false to say that if F is both
xrjne, with an n x n complex matrix E, then F = PE for some
n x n positive definite matrix P. We illustrate this with an example.
Example 4.1.
F- (,r2 [1
1
1
-11
1
[10
Let E = [01
0,
1
0
0] and F
1
[
-1
11)
1
1
1
Then
would be unitarily similar
to (= unitarily conjunctive with) E. So by Lemma 1. 5 F is both
/11-1
to, and _APlie with, E. Now suppose there is a positive
definite matrix P = [ y y] such that F = PE. Then
1
F
l
2l
=
x7
Z]
[10
00]
0
which is impossible.
[y 0]
Finally, we state without proof a result of Stein-Pfeffer
which may be expressed in the following form.
98
Theorem 4. 14. (Stein-Pfeffer, [13]). Let A be an n x n stable
matrix. Then an n x n hermitian matrix H is the hermitian part of
a matrix B Ott to A if and only if index (H) > the maximum
number of Jordan blocks at any one eigenvalue of A.
§4. 5.
Accretive Matrices
In this section we consider a special class of matrices called
accretive matrices, which are defined in §4. 4. We first state
without proof a lemma from [3].
Lemma 4.4. ([3], Lemma 5). If B is an n x n accretive
matrix, then B is conjunctive with a diagonal matrix
diag
(e1P1
e432,
,
e
n) with
Tr
2
t
2
> P> -
>
2
In [ 3], the authors gave a necessary and sufficient condition
for the existence of a lower triangular matrix conjunctive with an
accretive matrix. However, the proof was omitted in [ 3]. We
supply here a proof by Ballantine.
Theorem 4.15. ([3], Theorem 5).
Tr
z
>
13
>
>
.
.
Let
.
>p
>-
Tr
Then there exists a lower triangular n x n complex matrix T
conjunctive with diag (e iP1 ,
e i P-2 ,
,
eii311),
and having diagonal
99
entries t1
e
t
1,
2e
ia 2
,
tneian (in some order) with t1, t,
, tn
positive and
>a
Z> a
2
1
>
>a
Pk
al + a2 +
n
>
-
Tr
2
if and only if
(3.1 + P2 +
ak
for 0 < k < n, with equality holding for k c {0, n}.
Proof: The "only if" part follows from results proved in [3]
plus Facts 1. 5 and 1. 6. For the "if" part, we will rely mainly on
Facts 1. 5 and 1. 6, and Lemma 1.10 and use induction on n. We will
state the induction assertion, Pn, as a lemma.
Lemma 4. 5. (= induction assertion, Pn).
Whenever
>1
and
and
for all
7T
2
?'
132
Pn
>
12
>a >a
Pl + P2 +
k E {0, 1, 2,
P1 + P2 +
>
>a
P-->-
>
-
+ a2 -E.
Tr
2
ak
... , n} and
Pn =
+a2 +
an'
then there is a lower triangular matrix T conjunctive with
100
diag (el,
e.432,
and having diagonal entries
,
eial eia2
e
ian in this order.
Proof of Lemma 4. 5.
P1
is trivially true, so suppose n> 2
Let pi., i3', pn, a 1, az, ..., a n be as given
and Pn-1 is true.
in the hypothesis of Pn and let
D = diag
Note that
{i
{1, 2,
a1 >
1
n-1
2
... , n-1} :
(ei131, el-132,
+ a)n >
(p
2
n-1
,
+ p ),
e'').
so the set
(pi + pi+1) } is nonempty. Let j be the
ai >
minimum of this set.
Case 1. j = 1.
Here a
D[1, 2] =
so we have a conjunctivity (see Lemma 1.10)
1
e1 0
eicel
L2p
with p = isin 2(131 - 132 -
Let 81 =
2
P2)
.
si31-
n
2
(
a1 -
0
e1(P1 + P2 - al)
Pi2 "2 )
,
(which is real because
(2 al- pi - p2) = 2 (pi - ai)_> 0).
2
= p1 + p2 - al , and 6. = p. for i > 3. Then (applying
the [1, 21 -subconjunctivity indicated by the above conjunctivity to D
gives us that) D is conjunctive with a lower triangular matrix B
16
i8)
i61
en
and such
whose diagonal entries are (in order) e, e
101
that B(1) is diagonal. Also,->
6
2
+ P2 - al
52 =
6
+
2
52 + 53
> 6>
>62 >
2
a2
a2
63
a3
+ (P1 + P2
5k =
>
+
al + (al
a2 +.
Pk)
ak)
= a2 + a3 +... + ak
for all k e {2,3,
n} and 62 + 63 +... +6n = a2 + a3 +... + an.
,
Thus, by Pn_1, B(1) is conjunctive with a lower triangular matrix
T1
of diagonal (e1a2
,
elan) and hence B itself is conjunctive
e1a3
with
T = reial
G
01
T1.1
which satisfies the conclusion of Pn.
Case 2: j > 1 and a. < 13.
.
J
Again we have (j < n and)
arg diag D[i, i+11 = (pi, R
D
(ay Pi 4- Pj+1
)
a.) and hence
B, a lower triangular matrix whose arg diag is (61, 62, .
with 63. = a 3., 6i41
Pj
aj '
P
P j --I- 1
and 6. = 13. for i
{j, j+1}, and the only (possible) nonzero
,6)
102
off-diagonal entry of 13 is
permuting -1- 1, 2,
Now
.
B+1
apply the cyclicly-
j] -subconjunctivity to B and get a lower
,
triangular matrix
e
c=
of arg diag =
y2,
0
LG
B(j)
= (a),
5
8
,/.8
.
2'
) with
C(1) = B(j) diagonal. Also
Nj
T
y 3+.
2
+ P2 +"
=
+
a1
Y2 + Y3
Yk
Pk-1
+... + ak-1 for k < j
a2
Pj-1)
= (P1 +P2 +
3+2
(3*
"."
2,
T
:3
Then2 Cr-2> 3°
>>
T
n)
>
.
+. .
A
= (al, a2, ... , aj ,
n
>
-11-- and -1.--r> T
2
2
+ .yk> Tz
Y 2 + Y3
+ Yn = (P1 + P2
+
. +k
+...
a1+a2-F...
3
> Tn> -
>T>
E
.=
2
and)
{2, 3... , }. and
aj
+ Pn)
A
+oz.+
for k > j
, an).
T3 +... + Tk for k
N2 +
aj)
Pk) - ai
+a2 +... +
(T
j+1
+ Pk)
(P11 + P2
Let
(Pj
,
.. +an
103
=
T2 + T3
Tn.
Tht.s we can apply Pn-1 to C(1) and get it conjunctive with a lower
triangular matrix T2 of arg diag = (T2, T3,
Tri) =
(
ae
A
a.,
an).
Applying to C the corresponding
(1) -subconjunctivity, we get
T=
e
G1
0
T2_,
By successively applying the [1, 2], [2,3], ...,
j]
subconjunctivities given by Lemma 1.11 (for the [1, 2],
... , [ j-1, j]
principal submatrices, respectively), we get a matrix conjunctive
with T which satisfies the conclusion of Pn.
Case 3. (j > 1 and ) a. >
J
J
Here, by the minimality of j, we have a.j-1
<
2
j-1 +p.),
j
SO
a,j <(pJ-1 + 13.). Thus
3
arg diag
D
j-1, j] = (pi_i, 13i) --->
+ pi - ai, ai) and hence
B, a lower triangular matrix of
arg diag = (Sr 82, .
Pj+1'
(Pr
Pj-1
Pj
aj'
Pn)
and such that the only nonzero off-diagonal entry ofB is
J, J- 1
he
104
rest is pretty much as in Case 2,
n(J)
C=
B
ia
Arg diag C = eyl
.
e3
LG
by the cyclic-permuting
0
n] -subconjunctivity.
[j, j+1,
'Z' ..., yn) =(61,
..., on, ai) with
,
C(n) = B(j) diagonal. Also
Y1 + Y2 +
+ Yk = P1 +132 +
>a1 +a2
+ Y2 +
+...
+ak for k < j-1 ,
Pj-2) + (Pj-1 + P.
Yk = 431 +P2 +
1 +a2 +... +a.
j-1
Y1 + Y
+
+
Yk = (P1 +
f rk=j-1, and
o>a
(P j-1
(13j-i +
Pk+1)
= (31 + P2 +
Pk+1)
cri + a2 +
aj)
C\r. +
+ Pj
aj)
aj
+ ak+i
for n> k > j (and equality holds for
k = n-1).
Thus let (T1, I 2'
Tn-1) = (al,
aj'
an).
105
Then y + y
+ yk
+.
for k E
+ T 2 +.
T1
.
n-1)
with eqqality for k = n-1.
Also
Tr
T
2
1r..
2 >
n-1
>
>
and
2
1n-l>
Yj-2 = Pj-2> 13j-1> 15j-1
=01.
Thus, by P1, C(n)
Tr
Tr
/'2
2
>-
=
(aj
,
because
13j) = 6j-1 = Yj-1
+ [3.) - a.> a.> 13.> f3.
+1
=y
B(j) is conjunctive with T2, lower triangular
A
with arg diag = (T1,
) = (a
j
an).
Apply to C the corresponding (n) -subconjunctivity gives us
[ T2
T=
0
ia.
which is again conjunctive with, by
G1
repeated application of Lemma 1.11, a lower triangular matrix
satisfying the conclusion of P.
We invite the readers to compare Theorem 4.12 and the
following result, Theorem 4.16.
Theorem 4.16. Let B be an n x n accretive matrix and A be
an n x n complex matrix. If A is
with B, then A is stable.
Proof: Let B be an n x n accretive matrix and A be an n x n
complex matrix. Suppose A is
With B.
Then there exist
106
nonsingular matrices C and D such that C-1AC= D* BD. Since B is
accretive, B + B is positive definite.
*
*
*
Now
*
D BD + (D BD) = D BD +D B D=D (B + B )D
is also positive definite, so D BD is also accretive. Since every
accretive matrix is a stable matrix,
C-1AC
is stable and hence so
is A.
Let B = diag (e
Theorem 4.17.
2
>p >-.2
>p >p 2
with eigenvalues tle
positive and jr- >
2
Then
a1
ia1ia
t2
e
>
Let A be an n x n complex matrix
2
tn e
+ a2 + .
+13k >
131 +132 +
ian with t
.
,
t2'
t
n
> a> - Tr. Let A be 14,4 with B.
.
2
eiPn) with
e
I,
.
.
+ ak
for all k
{1, 2,
.
and equality holds for k = n.
Let B = diag (e .431 ,
Proof:
Tr
2
>p>p >.
.
.
>
1
with eigenvalues
pn
ia 1
t1 e
t2e
1(32
,
,
e
ipn ) with
Let A be an n x n complex matrix
> - Z.
ia
,
e
2,
... ,
tne
ian with
t1,
t2, ..., tn
>Let A be ilfra6 with B.
2
> an
Then there exist nonsingular matrices C and D such that
positive and IT- > a1
2
&2
a2
>
n}
107
C
-1 AC = D * BD. Then by Schur's Triangularization Theorem there
exists a unitary matrix U such that
*
U-1 (C-1AC) U = U (D BD) U = T
is a lower triangular matrix with
ia
ia
diag T = ( t le
1,
ian
ten).
... ,
2,
t2e
Since T is conjunctive
with B, by Theorem 4.15
+
for all k
E
Pk
P2 +
{1, 2,
al + a2 +
ak
... , n} and equality holds for k = n.
We could extend the following result, Theorem 4.18, in the
obvious way to the case where A is not assumed stable. The
increased intricacy needed to word this extension would however
tend to obscure the content, so we satisfy ourselves with the statement below.
Theorem 4.18.
2
>
> 13
1
p2
>.
. .
>p
Let B = diag (e
11.
ia
positive and
2> a1
>
a2
ia
1,
>
..
e
Pi131
2
,
e
n) with
Let A be an n x n complex matrix with
> - -2.
distinct eigenvalues tle
,
t2e
.
>
2'
n
... ,
> - 3-r-.
ix n with
.
t
tne
Then A is
t
2'
with B
108
k E{i.,
.
,
.
+ ak for all
+ a2 +
+pk >
if and only if pi +132 +
n} with equality holding for k = n.
Proof: The "only if" part follows from Theorem 4.17.
("If").
2
i132
Suppose B = diag (e
> pn > -
>p >p
1 1_2_>
e
,
ia
with distinct eigenvalues tle
tl' t
,
t positive and
>
2,
t2e
a1
) with
>
a2
,
tne
ian with
> a> -
>
+ ak for all k E {1, 2, ...n} and
+ a2 +
+13k>
(31 + [32 +
iP n
ia
1,
2
e
,
Suppose A is an n x n complex matrix
.
2
,
equality holds for k = n, then by Theorem 4.15 there exists a lower
triangular n x n complex matrix T conjunctive with B and having
al
diagonal entries tie ia1
eigenvalues tie
ia 1
t2e
,
2e
ian
ia2
tne
ia 2
ian , A is similar to T and so
tne
,
.
Since A has distinct
.
A is Ike" with B.
ip
Theorem 4.19.
>P1 > P2 ?-
with t
Pn
t' tn
2'
Let B = diag (e
>-
2
.
,
e
LetA = (tie, t2e
positive and
Tr
>
1> a2
,
,
e
ipn
) with
ia
ia2
.
>
a
>-
tea)
,
2
109
be the set of all n x n matrices similar to lower triangular
Let
matrices of diag = A
.
Then there exists A E OA which is
with B if and only if pi + p2 +
.
al
Pek
a2 +
for all k E {1, 2, ... n} with equality holding for k = n.
Proof: The proof is a slight modification of the proof of
Theorem 4.18.
Theorem 4.20. Let
r3
.
r+1' 3r+2'
, (3n
t
(i)
(32,
2
,
and B be conjunctive with
pr by 13 and
by pl. Let A have eigenvalues
1
ia2
(tie
> (3 >p, > -
(Denote (3
.
n- r
2
,
t2e
.
.
,
te
t2" tn positive.
4. (kf)
for all k E
) with
2
> a.i > a2 > .
.
.
> a > -2 and
Let A be .,141..e.. with B. Then
a2 +
{1, 2,
..., n) and
and equality holds for k = n;
ia.
A has only linear elementary divisors at t.e
if a.
p or = p.17
at each eigenvalue of sgn
ei(3,
A has no more than n - r
110
elementary divisors, or at each eigenvalue of sgn
e
ipt
,
A
has no more than r elementary divisors.
Proof: Let
>
2
p> p' > -
2
and B be conjunctive with
ia
ia2
eipr 9
with
Tr
>
Let A have eigenvalues (tle
i
eip In-r .
> a2 > .
.
.
A be xidne with B.
> a> -
Tr
2
and t
t
$
, ...,tne
t2 e
Let
tn positive.
2'
n)
Then
follows immediately from Theorem 4.17.
Let a = p. Since p >
that % = a2 =
tl' t2' ..
t2
= p.
=
then rp + (j-r) p,' <
>
a1 + a2
>
.
. .
> a. > .
.
>
an
it follows
j > r,
We also have j < r.ifFor,
contradicting (i). Since
. . .
tn are positive, it suffices to assume
=
1.
Hence diag T[1, 2, ..., j] = (e
(3,
e
ei43),
13,
(same matrix T conjunctive with B as in Theorem 4.17). We
claim that T [1, 2, ..., j] = diag
(eiP,
suffices to show that T[p,q] = diag
(eiP, eiP) ,
ern).
It
1<p<q<j
For if T [p,q] is nondiagonal lower triangular, say
.
111
eip
0
-2y
11
with y >0, then by Lemma 1.10 there is a
conjunctivity having the effect
[1
e16
0
eip
eip
2y
L 2,4 2
1
for all real 6 >0 such that
0
I sin 6
_
sin26
.
and
<
42
sin2 + cos 6 >0.
Thus, by Fact 1. 4, ITT) contains ei(P+6) Since ei(P+6) is outside the
r(B) contradicting
sector r (B) for all sufficiently small 5 > 0, r(T)
the fact that T is conjunctive with B. Thus T [p,q] must be a
Now, since A is similar to T, A has only linear
i«
divisors at t.e
if ai = 13. Similarly, if a -= P , then
diagonal matrix.
e
Jlementary
J
a. = a.
=
. . .
=
.
.
= pi and r < j < n as al. > a2 > ... > a > pt. Thus
an
by the same argument we can show that
T [ j, j+1,
J
.
,
n] = diag (eiPl,
and
eiPI ,
.
I
e143 )
,
J
if a. = pi.
J
(iii). To show that A has no more than n - r elementary divisors
at each eigenvalue of sgn
eiP,
it suffices to show that T has
the same property. Since
- ieiPB* = Or
=
Or
-i(eiO3-(31) - e-ig3-131))
9 [ 2 sin (p-p1)] In-r
In-r
112
and sin (13-
p'
.
> 0,
-
.
B-
B
*
is a matrix of rank = n - r.
r and T has m elementary divisors at some
Now suppose m> n
eigenvalue teia of T of arg = a, with 3> a>
and t > 0. For the
sake of convenience, let X = teia. (We denote nsp (M) as null
space of the matrix M). So the number of elementary divisors of T
at X
= dim (nsp (T - X I)
= dim (nsp (T
)
)
= m.
*
be an orthonormal basis for nsp (T --RI) and
m
Let xl' x2'
xl, x2, ..., xm, xm+i, ..., xn be an orthonormal basis for Cnx1 .
Let U = [x1 x2 ... xn] = [U1 U2] , where U1 = [x1 x2 .
X]
m
is
n x m. We note that U is unitary,
UU
1
[U,U2] =
U2 U1
U2
Thus U T U
U1 U1
[Ul* -
U1 U2
Im
U2 U2
0
T [Ui U2]
U2*
U,
U2
[T 'U1 T U2]
U,
[T
U,
U1
T U2]
I=i
In-m
113
T1
X
ww10
Im
S*
0
X
T1
Im
Hence U TU
=S
T1
where G
S[m+1, m+2,
.
,
n
I
1, 2, ..., m]
and
T1 = S[m+1, m+2,
w".
Thus ie-iPS - ieil3S*
2 sin (B-a) Im
GieI -ie
=
.
1
Since 0 <
- a< Tr, sin (13-
> 0.
So the rank of ie-iPS
must be at least m. Since m > n-r and the rank of ie-if3B
is n-r,
.ip
S - ie
iR
ieiPS*
iei1313*
cannot be conjunctive with ie PB-ie PB *
Hence S cannot be conjunctive with B. For if S were conjunctive with
B, then there would exist a nonsingular matrix D such that
D BD = S, and thus we would have
.
-ipS - ie ipS
.
.
=
(D BD)
ie
=
(D *BD)
i.eie
iB
(D BD)
(DBD)
*
114
=D
(ie
- ieipB*) D
hence would be conjunctive with le -ip B - ieipB*. Therefore T can.
not be conjunctive with B, this is a contradiction. Hence A has no
more than n-r elementary divisors at each eigenvalue of sgn
eir3
Similarly, we can show that A has no more than r elementary
divisors at each eigenvalue of sgn
e'.
We summarize our main results in Chapters III and IV in
the following Tables 3, 4 and 5.
.
Table 3.
n x n complex matrices.
B is CAtri to A
A
nonzero scalar
complex
complex
unide finite
> A B E7-1
(trB ) A
Reference
B`eA
Theorem 4.1
2
Theorem 4. 2
no
Theorem 3.5
Real diagonal,
positive distinct
complex
BE
nonsingular,
nonsingular
cohermitian
B =Band
nonsingular
{(A, B), (-A, - B))CP x ciD
3
ch. roots
similar to real
Theorem 3.6
sig B = sig A
diagonal matrix
Nonsingular,
real diagonal,
distinct ch. roots
nonsingular,
nonsingular
contrahermitian
nonsingular,
nonsingular
contradefinite
nonzero ch. roots
are distinct, has
only linear elem.
divisors at 0.
Theorem 3.7
codefinite
with ch. roots
real and distinct
distinct ch. roots
3
contrahermitian
-{(A, B), (-A, -B)} C
p2 x p3
Corollary 3. 7.1
codefinite
-irregular
sgn (det A) = sgn (det B)
<=> rank B = rank A
Theorem 4. 3
Theorem 4. 5
Table 3. Continued.
A
B
nonzero ch. roots
,X
are
X 1,X 2,
-regular
contradefinite
distinct, has only
linear elem. divisors
B is efriZ to A
Reference
rank B r and
(B) = sgn (X IX 2...X r)
n
Fact 4.1
at 0
idempotent of
rank r,
complex
0< r< n
0
a
9 13
ED
I
On-r-s, Theorem 3. 8
S
for s < r
[a
s
is-is air
ol
[Is
In-r
complex
ED a I
1
13, 0< r < n
y
r-s
ED
13
In -r-s
13_1
Theorem 4. 4
s < min {r,n-r}
Yy
r 2' "
Ys
Positive
nilpotent
rank = n-1
complex
Brestrictly lower triang.
matrix of rank n-1
nonderogatory,
with single ch.
root X
0, n > 2
complex
B?dlower-triang. nondiagonal
Theorem 4. 6
matrix with diag= (X , , ...X )
B EX p 3 but B N.p
Theorem 3.12
Table 3. Continued.
B is CAut, to A
A
Reference
normal
hermitian
A'eB
Theorem 3. 4
complex
hermitian
A oehermitian matrix of
same rank and
signature as B
Theorem 3. 3
unitary
nonsingular
<
hermitian
involutory
> A is a symmetry and
Theorem 4. 9
index A = index B
<=fr A'ZIB
unitary
B is a symmetry and
Theorem 4.10
index B = index A
<=4 13/A
accretive
complex
A is stable
with distinct
ch.ia
lai
roots(tle )...,
t1, t2,
tea),
, tn positive,
>a>a>...>a >- 2
7r
Tr
diag. (e
IT
>@_>B
>
2- rir2
Theorem 4.16
ip
i@
, ... , e
>a, >--Tr
2
n)
#=?
Pi+1324--
-hk?-0i+a2+.
for all k E {1, 2,
+Cek
... , n}
and equality holds for k=n
Theorem 4.18
Table
Continued.
B is
A
stable
c
er
e e ip' in-r,
Tr
2
C/frj,1
to A
Referen.c,,
A has only linear elem.
divisors at teia
if a = p or = pi
i(1)
(2) At each ch. root of
A has no more
than n-r elem. divisors,
or at each ch. root of
sgn
eip,
sgn
e', A has
more thanr elem.
divisors.
no
Theorem 4. 20
Table 4. Matrices similar to triangular matrices.(Let
(x'
,
2
n
n x n nonsingular matrices similar to triangular matrices of diag = A
a,_
complex
X.>0, i--=-1, 2, ... , n
1
X
1
i
--,
\{0},
1, 2,
<=> B e cp3
nonsingular
cohermitian
... , n
,(
nonsingular
x. e GR\{0},
i = 1, 2,
B is CM/ to some A E
B
.(=;,
contrahermitian
... , n
,
1j31ii3n
code finite
be the set of all
) and
. )
Reference
Theorem 3. 9
B is hermitian,
A is diagonable, and
sig B = sig A
Theorem 3.10
either sig A = n and BEP 3
or else sigh= -n and B
Theorem 3.11
ia.
X
= t.e
i
t
i=1, 2, ...
Tr
2 >c11>a2.
,
t.>0,
1
diag(e
, ... e
),
,n,
<=> Pi+P)2+.
+Pikai+az+.
for all k e {1, 2,
Tr
>an>-2
Tr
>3 >.
2 F1-1-2
Tr
>f3 >-n
2
... ,n}
and equality holds for k = n.
Theorem 4. 19
Table 5. Miscellaneous results.
A
Conditions
Reference
nonsingular
There is a unitary matrix B such that B is both
CA.-a, to, and 4111/C with, A
Theorem 4. 8
stable
There is an accretive matrix B such that B
is both CALL to, and _14/2C, with, A
Theorem 4. 12
stable
Hermitian matrix H is the hermitian part of a
Theorem 4.14
matrix B Cm,L to A
< > index H > max. no. of Jordan blocks
at any one eigenvalue of A
121
BIBLIOGRAPHY
C. S. Ballantine, Triangularizing Matrices by Congruence,
Linear Algebra and Its Applications, 1(1968), 261-280.
C. S. Ballantine, Products of Positive Definite Matrices.
Linear Algebra and its Applications, 3(1970), 79-114.
IV,
C. S. Ballantine and C. R. Johnson, Accretive Matrix Products, Linear and Multilinear Algebra (to appear).
S. Barnett and C. Storey, Analysis and Synthesis of Stability
Matrices, J. of Differential Equations, 3(1967), 414-422.
M. -D. Choi, Adjunction and Inversion of Invertible HilbertSpace Operators, Indiana University Mathematics Journal,
Vol. 23, 5(1973), 413-419.
A. W. Gillies, The Physical Content of Autonne's Lemma,
Siam J. Applied Mathematics, Vol. 19, 1(1970), 142-143.
K. Hoffman and R. Kunze, Linear Algebra, 2nd ed. Prentice
Hall, 1971.
D. G. Hook, Effects of Conjunctivity on the Inertia of Complex
Matrices, Ph. D. Thesis, Oregon State University, 1974.
I.
J. Katz and M. H. Pearl, On EPr and Normal EPr Matrices,
Journal of Research of the National Bureau of Standards (1966),
47-77.
M. Marcus and H. Minc, A Survey of Matrix Theory and
Matrix Inequalities, Allyn and Bacon, Boston, 1964.
S. Perlis, Theory of Matrices, Addison-Wesley, Reading,
Mass.
H.d
n
2
1958.
Schneider and G. P. Barker, Matrices and Linear Algebra
ed. , Holt, Rinehart and Winston, Inc., New York, 1973.
P. Stein and A. Pfeffer, On the Ranges of Two Functions of
Positive Definite Matrices II, ICC Bulletin, 1967, vol. 6,
81-86.
122
14.
0. Taussky, A. Remark on a Theorem of Lyapunov, J. Math.
Anal. and Appl. 2(1961), 105-107.
APPENDIX
Page Index to Lemmas, Theorems, Corollaries, Facts,
Table
1, p.
2, p.
Definitions, Examples and Tables
Theorem 1. 1, p. 10 Theorem 4. 11, p.96
4. 12, p.96
1. 2,
p.11
61
62
3, p. 115
4, p. 119
5, p. 120
Fact 1. 1,
p.
1. 2, p.
1. 3,
1. 4,
1. 5,
1. 6,
Fact 4. 1,
Lemma 1.
5
6
6
p.
p. 7
p. 9
Theorem
p. 10
p. 84
1, p.
1. 2, p.
1. 3,
1. 4,
1. 5,
1. 6,
2
2
p. 2
p.
2
p.
3
p.
1. 7, p.
3
1. 8, p.
1. 9, p.
123
p. 11
p. 13
p. 14
p. 14
p. 14
p. 14
I. 16, p. 15
Lemma 3. 1, p.67
Lemma 4. 1, p.85
4. 2, p. 87
4. 3, p. 89
4. 4, p.98
4. 5, p.99
1. 7,
p. 15
p. 15
p. 16
1. 8,
p.16
2.
2.
2.
2.
2.
2.
2.
p. 19
p. 19
p. 21
p. 21
Theorem 3.
1,
2,
3,
4,
5,
6,
7,
1,
3.
3.
3.
3.
3.
3.
2,
3,
4,
5,
6,
7,
3. 8,
3. 9 ,
p.23
p. 24
p.36
p.64
p.64
p.65
p.65
p. 65
p. 66
p.68
p.70
p. 74
3. 10, p. 74
3. 11, p. 75
3. 12, p. 75
2. 2, p. 22
2. 3, p. 26
2. 4, p. 26
2. 5, p. 27
2. 9, p. 35
1. 5,
1. 6,
p.37
p.40
2. 10, p.42
2. 11, p.45
2. 12, p.54
2. 13, p.55
Lemma 2. 1, p. 20
2. 6, p. 27
2. 7, p. 30
2. 8, p.32
p.15
p.15
2. 8,
2. 9,
4
4
4
1. 10,
1. 11,
1. 12,
1. 13,
1. 14,
1. 15,
1. 3,
1. 4,
Theorem
4.
4.
4.
4.
4.
4.
4.
1,
2,
3,
4,
5,
6,
7,
4 . 8,
p. 77
p. 77
p. 78
p. 78
p. 83
p.91
p.93
p. 94
p.94
4. 10, p.95
4. 4 9,
4. 13, p. 9 7
4. 14, p.98
4. 15, p.98
4. 16, p. 105
4. 17, p. 106
4. 18, p.107
4. 19, p. 108
4. 20, p. 109
Sublemma 4.3. 1, p. 89
Corollary 2. 3. 1, p. 21
Corollary 2. 7. 1, p. 36
2. 8. 1, p.38
2.8. 2, p.39
2.9. 1, p. 41
2. 10. 1, p.45
2. 11. 1, p.50
2. 12. 1, p.54
2. 13. 1, p.57
Corollary 3. 7. 1, p.69
Corollary 4. 4. 1, p. 82
Definition 1. 1, p. 2
1. 2, p. 2
Example 1. 1, p.3
Example 4. 1, p. 9 7