AN ABSTRACT OF THE THESIS OF
for the
RICHARD DAVID HILL
(Name)
in
Doctor of Philosophy
(Degree)
\\\C
presented on
Mathematics
(Major)
Title: GENERALIZED INERTIA THEORY FOR COMPLEX
MATRICES
Redacted for privacy
Abstract approved:
(David Carlson)
The complex matrices
Al' A2'
As
quasi-commutative iff each of Al' A2'
A A. - A.A. (i, j = 1, 2,
s).
are said to be
commutes with
J
Let Al' A2'
be quasi-commutative complex
As
matrices of order n whose eigenvalues under a natural correspondence are
X
X
k
(s)
k
be Hermitian of order
k
'
(k = 1, 2,
n) and let
There exists a Hermitian H of order
s.
dijA,HA.
> 0 (positive definite) iff
1
such that
j=1
(k = 1, 2,
dijXk(i)TCk (.93
j7-1
,n). There exists an H > 0 such that
d..A.HA. > 0
13 1
3
Moreover, if
dijXk(i)X(i)
>0
k
iff
j=1
(k = 1, 2,
j=1
Tr(D) < 1
D = (d..)
and
v(D) < 1
then whenever
,n
0
d..A.HA.
> 0,
3
InIA 1 , A2,
In H
1.3
,
As ,D} where
j=1
In {A1' A2'
where
Tr,
is defined to be the ordered triple
As,D}
(Tr, v, 6)
v, and 5 are the numbers of positive, negative, and
zero values respectively of
d .X
X
i3 k
(k = 1, 2,
k
,n).
If
j=1
Tr(D) > 1
or if v(D) > 1,
matrices A1, A
2,
and n> 2, there exist complex
of order
, As
n
which commute in pairs
(and thus are quasi-commutative) and a Hermitian H
>0
d..A.HA.
13
such that
and
3
I.
In H
In{A , A2,
of order n
As, D} .
j=1
A similar body of theory is developed for
fi(A)Hgi(A )
i= 1
(in the role of
d..A.HA.
13
1
3
)
where A is a complex matrix of order
j=1
n; H is a Hermitian matrix of order n;
tions defined on A; and g
g
1'
,f m are matric func -
are matric functions defined on A.
Both sets of results generalize the theorems of Lyapunov
and Stein and the Ostrowski-Schneider-Taussky Main Inertia Theorem.
Specializations of the previous results may be thought of in
terms of exclusion or inclusion regions for the eigenvalues of the
matrices involved. The final results are ten of these analytic
geometry applications.
GENERALIZED INERTIA THEORY FOR COMPLEX MATRICES
by
Richard David Hill
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Doctor of Philosophy
June 1968
APPROVED:
Redacted for privacy
Associate Professor of Mathematics
In Charge of Major
Redacted for privacy
Chairman of Department of Matherhatics
Redacted for privacy
Dean of Graduate School
Date thesis is presented
April 11,
1968
Typed by Carol Baker for Richard David Hill
ACKNOWLEDGMENTS
I wish to express my gratitude to Professor David Carlson
for his crucial part in the development of my ability to do research
in matrix theory and thank him for the time and energy which he has
so graciously expended in my behalf.
I wish to thank Professors Anselone, Arnold, Brown, Carlson,
Lonseth, Poole, Stalley, and Stone for their contributions to my
graduate education.
I am indebted to my wife, Willetta, for her encouragement
and patience as a graduate student's wife, particularly through the
research for and writing of this thesis, and to my parents for their
academic encouragement throughout my boyhood.
TABLE OF CONTENTS
Chapter
Page
INTRODUCTION
1
THE THEORY FOR
9
A1' A2'
s
THE THEORY FOR GENERAL MATRIC FUNCTIONS
41
SOME ANALYTIC GEOMETRY APPLICATIONS
62
BIBLIOGRAPHY
74
GENERALIZED INERTIA THEORY FOR COMPLEX MATRICES
CHAPTER 1. INTRODUCTION
The inertia of a complex matrix A is defined to be the
ordered triple. In A = (Tr, v, 5) where
values of A with positive real part,
real part, and
5
Tr
v
is the number of eigen-
the number with negative
the number with zero real part. The two classi-
cal theorems involving matrix inertias are those credited to Sylvester
and Lyapunov. The theory developed through generalizations of the
Lyapunov theorem form that branch of matrix theory commonly called
inertia theory.
Sylvester's Theorem: If P is non-singular and H is Hermitian,
then
In H
In PHP
.
Lyapunovls Theorem:
If A is a complex matrix, then there exists
an H > 0 (i. e., positive definite) such that AH + HA* > 0 iff
In A = (n, 0, 0).
These results were generalized independently by Taussky [9,10]
and Ostrowski-Schneider [6] to the following theorem:
Main Inertia Theorem:
Given a complex matrix A,
a Hermitian H such that AH + HA > 0 iff
there exists
6(A) = 0
(i. e.,
A
has no pure imaginary eigenvalues). Furthermore, if AH+ HA > 0,
2
then
In A = In H.
Matrices
A with lirn AO appear in many iteration
n-4-00
processes. Stein [8] characterized these matrices in the following
theorem:
Stein' s Theorem: The complex matrix A
satisfies lim An = 0
00
iff there exists an H> 0 such that H - AHA > 0.
The condition that
that
Ix kl <1 (k = 1, 2,
lirn An = 0
n)
where
is equivalent to the condition
X 1, X 2,
X
are the
eigenvalues of A. Householder and Varga observed a connection
between the theorems of Stein and Lyapunov. Taussky [11] showed
the equivalence of the Stein and Lyapunov theorems with a linear
fractional substitution. Independently, the author [3] showed that a
generalization of the Stein theorem was equivalent to the Ostrowski-
Schneider theorem. Furthermore, the author [3] proved equivalent
theorems corresponding to polynomials other than that of Lyapunov,
AH + HA, and Stein, H
- AFIA
Three basic generalizations of these theorems appear in this
thesis. They take the form of the Lyapunov theorem with H
positive definite and the two parts of the Main Inertia Theorem with
the much weaker condition of H Hermitian.
Since the two parts of the Main Inertia Theorem generalize
3
differently, it will be necessary to distinguish between them. We
shall call the first part the "iff" part and the second the "inertial"
part.
A 1965 paper by Schneider [7] motivated a deeper investigation.
The inertia theorem of this paper may be stated as follows:
Schneider' s Theorem: Let A, C1, C2,
matrices of order
be complex
Cs
which can be simultaneously triangulated.
n
Let the eigenvalues of A, Ck under a natural correspondence be
a.
y
(k)
,
i
n
1, 2,
and
For Hermitian H let
k = 1, 2,
T(H) = AHA
-Ck HCk
.
k=1
Then the following are equivalent:
lail2-
yi(k.)12
>0
(
i
1, 2,
n)
.
k=1
There exists an H > 0 such that
T(1-1)'> 0.
For all K> 0, there exists a unique H > 0 such that
T(H) =K.
The theorems of Stein and Lyapunov are corollaries of this
result. We now ask whether a similar theorem can be proven for a
Hermitian (instead of positive definite) H.
Can Schneiders
T(H)
be further generalized? Schneider gives a counterexample to any
sort of inertial generalization for his
A1,
Which functions of
T(H).
As, and H admit such a generalization?
The author's first results were for polynomials of the form
As, Al
p. (A)H qi(A*)
i= 1
where
P1'
and
q
q
were themselves polynomials.
In Chapter 3 we extend most of these results to
f.( )H g (A*)
1=1
where fl'
are matric functions.
fm and gl,
gm
The most profound results of this thesis are found in Chapter
2 where the theory is developed for a set A1, A2,
As
of com-
plex matrices.
For this development, we need to consider conditions which
will enable us to compare the eigpnvalues of a set Al, A2,
of complex matrices of order n. Classically, A1, A2,
, As
As
are said to have the Frobenius property if there exists an ordering
of their eigenvalues, say Xk(1) , X.k(2)
such that any polynomial
p(Al, A2,
(s)
Xk
As)
(k= I, 2,
has eigenvalues
n(x (1),x (2)
k
k
x- k(s)) (k= 1, 2,
n).
The complex matrices A1, A2,
,
As
are said to be
simultaneously triangulable iff there exists a non-singular
S
such
that
SA.S.-
1
is (upper) triangular
s).
(j = 1, 2,
Simultaneous
3
triangulability and the Frobenius property are equivalent. In fact,
a theorem of Drazin, Dungey and Gruenberg [2] may be extended to
six equivalent conditions including these two and one which occurs
naturally in the development of this thesis but does not seem to ap-
pear in the literature. This is done at the end of Chapter 2.
At the point of Proposition 2.4 we need a slightly stronger
condition than simultaneous triangulability. The complex matrices
A1, A2,
A1, A2
are said to be quasi-commutative iff each of
As
(i, j = 1, 2,
commutes with A.A.3 - A.A.
3
A s,
still stronger condition on A
As
,
s).
A
is that they commute
in pairs.
We now summarize our major results. Let A1, Az,
, As
be quasi-commutative complex matrices of order n whose eigenvalues under a natural correspondence are
(k= 1, 2,
(s)
,,kk
D be Hermitian of order s.
and let
n)
(1) (2)
Xk
,Xk
We show
that a necessary and sufficient condition that there exist a Hermitian
H
of order
such that
n
d..
A.HA. > 0
13
1
is that
3
j=1
d
j=1
ij.
(i)X (j)
k
k
0
(
= 1, 2, '
11).
6
Furthermore, a necessary and sufficient condition that there exist
an H > 0 such that
d..
A.HA. > 0
13
j
1
j=1
is that
d.. X
(j) > 0
k
k
(k = 1, 2,
n).
j=1
In quest of the natural inertial generalization with Hermitian
H,
we define
In {A1,
As, 13}
to be the ordered triple
( Tr, v, 6)
of positive, negative, and zero values of
d.. X
If
A1'
A
As
X
k
(j)
k
(k
n).
1, 2,
are quasi-commutative,
d..
A.HA,
> 0,
13
1
j=1
and
D
is Hermitian with
that In H = In {A/, A2,
and
Tr (D) < 1
As, D}.
If
v(D) < 1,
11(D) > 1
we show
or if
v(D) > 1,
we show that there exist quasi-commutative matrices A A
of order n and a Hermitian
of order n such that
d..
A.H
A> 0
13
1 0 j
j=1
,
As
7
In Ho / In {A.1, A2,
and
As, D}.
The theory which is developed for
f.( A) Hgi(A*)
i=1
in Chapter 3 follows to a large degree the pattern of the theory
developed for
ij
Ai,HA
*
j=1
in Chapter 2.
Specializations of some of the results of Chapters 2 and 3
(especially the latter chapter) may be thought of in terms of exclu-
sion or inclusion regions for the eigenvalues of the matrices in-
volved. The analytic geometry of these results is particularly
interesting. We discuss this for some of them in Chapter 4.
We conclude this introductory chapter with a summary of the
notation to be used in this thesis. Some of this notation has already
been introduced.
Notation
A ;AI,
A
n X n complex matrices
n X n Hermitian matrix
A
A> 0
conjugate-transpose of A
A is positive definite
8
Re {A}
Im {A}
Tr (A)
the number of eigenvalues of A in the (open)
right half-plane
v(A)
the number of eigenvalues of A in the (open)
left half-plane
5(A)
the number of eigenvalues of A on the imaginary
axis
row vector
X
im {f(X )}
re {f(). )}
column vector
1
1
{f(X) - f(T)}
-2- {f(X)
f(). )}
direct sum
AkE)
A2
Kronecker product of A1 and A2 (cf. [4, p. 8])
CHAPTER 2. THE THEORY FOR Al' A2'
As
The results of this chapter answer the questions raised in
Schneider's paper [7, p. 15-16] as well as generalizing the results
of Lyapunov, Stein, Sylvester, and the Main Inertia Theorem.
Let
of order
(1)
X.
k
X
A1/ A2,
be quasi-commutative complex matrices
As
n whose eigenvalues under a natural correspondence are
(2)
k '
(n)
k
Hermitian of order
n)
(k = 1, 2,
and let
D= (d..) be
Theorem 2. 6 gives a necessary and suffi-
s.
cient condition that there exist a Hermitian H of order n such
that
A.HA.
> 0,
1
1
cl.
ij
j=1
namely that
(i) (j) / 0
k
ijk
j=1
d.. X
X
(k=1, 2,
n)
.
Theorem 2.7 gives a necessary and sufficient condition that there
exist a positive definite
13
namely that
such that
H
&HA.j > 0
1
j=1
d
j=1
ij
X
k(i)X(i)
k
>0
(k = 1, 2,
n)
.
10
Propositions 2.4 and 2.5 reveal more structure than the abovementioned theorems. Proposition 2. 5 can be proven with a simul-
taneous triangulability hypothesis on A1, A2,
As
whereas
Proposition 2. 4 cannot. It requires the stronger hypothesis of
quasi-cornmutativity on A1, A2,
A. Moreover, the structural
development of Propositions 2. 4 and 2. 5 shows us why we get the
analogous "iff" results of Theorems 2.6 and 2.7.
With H positive definite, our inertial result
In H = In {A1, A2,
A
,
D}
is simply a rewording of Theorem 2.7
since both In H and In {A1, A2,
As, D}
are equal to (n, 0? 0).
A more challenging question concerns the inertial generalization with
H Hermitian. In the author's development, Lemma 2.10 is the
crucial part. That H be non-singular is not only a necessary condition that In H = In {A1, A2,
As, D},
it is used directly in the
proof of this result.
It is amazing that the inertia of the matrix of coefficients
D = (d..)
of
13
5
d..ij A.HA,
j
j=1
should determine which of these polynomials in Al,
As,
*
,
possess the In H = In {A1,
, As, D}
result.
Al ' - A,,
00
2
If
-
D has at most one positive eigenvalue and at most one negative
eigenvalue, we prove this result. If not, we construct a general
11
counterexample to show that no such generalization can hold.
We now begin our development of the theory.
Lemma 2. 1. If
is Hermitian of order
D = (d..)
s,
then
d,. A.HA.
j
13
=1
1,
is Hermitian for all Hermitian H and all complex matrices
A' A 2'
A
s
of order n.
S
Proof:
Id..A.HA,
ij i j
A HA-,
7.
i, j=1
j=1
d
A HA
133
1
j=1
d.,
A.1 HA.j ]
13
j=1
d..
A.1 HA.j
13
Thus,
is Hermitian.
j=1
Definition 2. 2. Let
matrices of order
A1' Az'
n
As
be simultaneously triangulable
whose eigenvalues under a natural corres-
(1)
(2)
pondence are Xk
(k = 1, 2,
,Xk '
X(s)k
be a Hermitian matrix of order s. We define
n)
and let
D
12
Tr
to be the ordered triple
As, D}
In {A.1, A2,
is the number of positive values,
values, and
6
v
(Tr, v, 6)
where
the number of negative
the number of zero values of
(i)
k
k
(k =
n).
j=1
Before our definition of In {A1, A2,
can be
As, D}
meaningful we must show that
d(i)
ij k
0)
k
j=1
is real for k = 1, 2,
n. We let A. =
in Lemma 2. 1 above. Then, since
H= 1
d
ij
X
(i)
k
(i) (i= 1,2
k
D = (d..)
s)
and
is Hermitian,
0)
k
j=1
is real
(k
1, 2,
n).
Lemma 2. 3. If
Al' A2' ' ', As are simultaneously triangulable
complex matrices whose eigenvalues under a natural correspondence
are
X
k
X
k(2),
h(s)
k
there exists a non-singular
(k = 1, 2,
S
such that
then given
E>0
13
SA S-1=
where
lb (j)
Proof:
Since
b (j)
12
13
bi n
0
x (j)
2
(j)
b23
b2n
0
0
X
b
0
0
(j
i/
A1,
A
112
x(j)
t 23
(i)
t (i)
(j)
t 3n(i)
0
-
R6 = diag (1,
R61 = diag (1, 5,
.
t (j)
(j)
(j)
0
62,
(j)
n
such that
Q
13
X
J
Let
3n
are simultaneously triangulable,
s
2
QA.Q-1=
> i)
s;
1, 2,
A2'
3
X
there exists a non-singular
x
(j)
b (j)
x
o2
6n-1).
3
.
(j)
in
2n,
61-n) where
(j =1, 2,
6>0 .
Then
By computation we have that
14
6
X
-
-
R6(QAjQ1 )R1
o
=
(i)
t12
x (j)
2
0
0
we choose
where M =
ItLEW I
Then
max
i,/ =1, 2,
j=1, 2,
1 61 -it(j)1
<
It
161
Proposition 2.4.
Let A 1 ' A2'
plex matrices of order
(j)
6
such that
0
It(j)1
'
As
(j=1,2,s).
< 6 < min
i.e.,
be quasi-commutative com-
whose eigenvalues under a natural cor-
k " kX(s)
X(2)
and let
H
> 0,
dij.A.HA.
1
j=1
6r1-3t3nj)
3
S
n
(1)
k
Hermitian of order s,
If
2n
is non-singular and S-1 =Q-1RE.
we have our result.
as
bit(i)'
6/t(j)
(i)
611-2t (j)
23
<
Then
X
t (j)
, n;
s .
We let S = R6Q.
Renaming
respondence are
in
13
0
Now, given E > 0,
(j)
62t (j)
(k = 1, 2,
,n),
be Hermitian of order
then
D be
let
n.
15
0
(k = 1, 2,
n).
>0
and H> 0,
then
dij Xk Xk
j=
(ii)
d.. A. H A
If
131
d. Xk
X
ij
.
>0
k
(k = 1, 2,
,
j=1
Proof:
We prove (i) by contraposition. We assume that
Al' A
As
have eigenvalues
X
(1)
k
(2)
, Xk
,
X(s)
k
(under the
given natural correspondence) such that
d
X
(i)x (j) = 0
k
ij k
j=1
for some fixed k.
Now by [2, p. 225] there is a common eigen-
vector for each A. (j = 1, 2,
X
(1)
k
X(2)
k '
(s)
k
e. ,
corresponding to
s)
there exists a (row) eigenvector
such that v Aj= X k(i) v (j = 1, 2,
A .*v * = X. (j) v * (j = 1, 2,
s).
,
s).
This gives us that
We have that
16
"lc
d..A.HA.
] v*
13
1
**
v)
3
d . . (vA .) H (A .
13
3
j=1
1
j=1
d..(X (i)v)H
13
k
(Dv*
)
j=1
d
[
.
X
13
(i)X (j)] vH v*
k
k
j=1
= 0.
This is a contradiction to our hypothesis that
d..A.HA.
> 0.
13 1
3
j=1
See [4, p. 69]
.
For (ii),
Thus, we have proven (i).
let
X
(1)
k
X
(2)
k
(s)X
k
(k = 1, 2,
n)
be the
given natural correspondence of eigenvalues of Ai, A2,
Again by [ 2, p. 225]
for each
,
X
(1) X (2)
k ' k '
(s)
k
As.
(k = 1, 2,
there exists a (row) eigenvector vk such that vk A. = X k(Dv k
(k = 1, 2,
, n; j 1, 2,
s). By the above calculation we have
that
s
d..A.HA.
i
13
3
j=1
*
*
d
vk
*
v
H
v
.XX(j)]
k
k
k
i3 k
i, j=1
(k = 1, 2,
n).
n)
17
Since H> 0
and
d..A.H
A.
>0
j
1) 1
j=1
by hypothesis, the numbers
vkHvk
d.. A. H A.
ij 1
j
and
(k = 1, 2,
vk
n)
j=1
are positive. Thus, the numbers
dijX k(0X k(i)
(k = 1, 2,
j=1
must be positive.
I
The following example shows that the hypothesis of quasi-
commutativity cannot be relaxed to simultaneous triangulability in
Proposition 2.4.
01
Let Al
(0
0
)'
A2
00
(0 1),
10
H = (0 1),
and
D
Then
2
dij .A.HA.j
1
j=1
ll
= AA
AA
22
10
00
= (0 0) + (0 1)
=
(10 10) > 0 .
10
= (0
18
However,
2
(i)
d.XX
ij
1
1
=X
(1)X (1) + X (2)X (2) = 0 + 0 = 0
1
1
.
1
1
j=1
A1A2 -
with
),
We note that since A 1A2 = ( 00 10 ) and A2 A1 = ( 00
00
, 0 1,
A
= Al. This commutator does not commute
= 'o (34'
2A
Thus
A2.
A1
and A2 are not quasi-commutative.
We have seen that the condition of quasi-commutativity on
, As gives us an eigenvector structure sufficient for the
Al' A2'
proof of Proposition 2.4. Whereas we would like this result to hold
for simultaneously triangulable Ai, A2,
,A,
the above counter-
example shows that this is not the case. Using [2, p. 222] , we can
show that A1, A
-2, ,A
simultaneously triangulable and the other
hypotheses of Proposition 2.4 imply that
d.. X (1)X(j)
k
k
0
j=1
for some
k
(but not all
k)
(k = 1, 2,
,n).
There is a condition which is slightly more general and much
more complex than quasi-commutativity which is strong enough to
prove the eigenvector theorem which in turn gives us the result of
Proposition 2.4. Drazin [1] calls it property
A
l' A2'
,As
be any finite set
r(0)
Q.
Letting
of complex matrices of
19
_1k)
(k = 1, 2,
order n, we define the sets I-
inductively as
(k-1) -C (k- 1) A.
the set of all matrices of the form C(k) = A.0
i
1
where C(k-1)
(i = 1, 2,
preceding set
runs through all members of the
Every matrix C(k) of the sets
r(k- 1).
r(k)
If every kth
will be called a kth commutator of the set F(0).
is the zero matrix, then
commutator of the given set
r (0)
is said to have the property Qk of generalized quasi-commutativity
of the
order.
kth
iff there is an integer
Q
We note that
pairs whereas
Conditions
such that Qk holds.
says that A1,A2,
Q1
commute in
As
is our usual condition of quasi-commutativity.
Q2
give us the generalization. Under property
Q3, Q4,
for A1$ A2,
(1)
k
k
",As are said to have property
Drazin [1, P. 226] proves thatthere is a common eigenvector
Q,
X
Now A1 , A2,
'
X
(2)
k '
As
(s)
-
'
k
corresponding to the eigenvalues
for
k = 1, 2, -
This result is sufficient
,n.
for the proof of Proposition 2.4.
Proposition 2.5.
lable matrices of order
respondence ar e
D= (d..)
Xk(1)
n
whose eigenvalues under a natural corh(s)
(2) ,
be Hermitian of order
dijXk(1)X(j)
k
,
j=1
be simultaneously triangu-
Let Ai, A2,
0
(k = 1, 2,
,n)
and let
s.
(k
1, 2,
n),
then there
20
exists a Hermitian H0 of order
such that
n
d..A.H
A. > 0
0 j
13
1
(a)
j=
H0" '
and
In
(b)
(ii)
If
d
ij X k
.
A
{A.
=
X
>0
k
s
2
D}
(k = 1, 2,
.
n),
then there
j=1
exists an H0 > 0
of order
such that
n
d..A.H
A
13
1
>0
.
j=1
Proof:
Since
A
l' A2'
are given to be simultaneously
s
triangulable, by Lemma 2. 3 given E > 0 there exists a nonS
s
such that S 1A.S
=ingular
A. + T.
=
dA.
iag {X (j), X (i)
1
triangular with
X
2
t
I
(1)}
< E.
sk = sgn (
and
(j = 1, 2,
where
s)
J
J
T.j = (t. (i))
is strictly upper
For notational convenience we let
d.. X
k
0))
k
(k = 1, 2,
,n)
.
j=1
Since
us that
D
is Hermitian, the remark following Definition 2. 2 gives
21
d
x.
X
ij k
(i)
k
j=1
is real (k = 1, 2,
Thus,
n).
In {A
,A, D} is well-
1, A 2,
defined and g is Hermitian where we define g by
= diag Is 1, s2,
,
sn}
.
A
,
By this definition,
= In {A
In
(1)
A. =S -1 A.S
=Let
A.+ T ..
3
J
A2'
s
D}
.
*
* -1
Then A. = S A. S
= A.+ T
3
3
.
J
Now
A
*
d XHA,
=
d..(A.+
T.) g (A.+T.*)
13
1
J
j=1
1
J
j=1
dji [ A. n-ii.+(r.tr.A-..+A.firr.--+T.AT.*)]
j=1
A
H A.j + E
d..A.
ij 1
1, j=1
where all elements of E can be made arbitrarily small.
22
Now,
d..13 A.
A.
x,
1
j=1
= diag
d
.X
i3
(i)(j)
X
(i)
,
d13X2
1
1
j=1
= diag
j=1
d .X (i)-K(i) I
i3
1
dX
..
k
since
j=1
d..X
dijX 2i)X2WI
j=1
1
j=1
>0
d..Xx0)/
n
n
sn
'
X.2
(i) 4
k
0
X
n
n
}
j=1
(k = 1, 2,
,
n)
by hypothesis,
j=1
s
Since
is fixed and positive definite,
d., A. g 71-.
i)
)
1
d..A111Z.44>
13
j=1
I-
J
, j=1
for sufficiently small E > 0.
Thus,
S
13
1
ij
3
i, j=1
Since
d.
1-1.A.4. * =
(
Sd../k.
-1A.1 S) fi' (s* A.S
*3
i, j=1
S
is non-singular and S-1
=S
*-1
we apply Sylvester's
Law of Inertia to get
*
d.,
1j
A.1 S) H (S A. S
-1
3
j=1
d
i, jl
or
i3
A. SHS A,* > 0,
1
))S
*
>0
0
23
d..
A. H A.> 0 where
ij 1 0 j
S*.
Ho = S
j=1
Once again appealing to the Sylvester Theorem we have that
In H0 = In H.
(2)
Combining (1) and (2),
In H = In {A1, A2,
As , D}
To prove (ii), we assume that
d
.
(j) >
iX k(1)X k
0j
j=1
,n)
(k = 1, 2,
and follow the path of the proof of
sk
sgn
k
ij k
=1
(i).
Now
(k = 1, 2,
j=1
Thus,
*
H
In
and H0 = S H S > 0.
For this positive definite
H0 we have that
..
13
A. H0 A.j > 0
.
j=1
Theorem 2. 6.
of order
(1)
Xk
X2 ,
Xk
n
Let A l' A2'
As
be quasi-commutative matrices
whose eigenvalues under a natural correspondence are
Xk
(k = 1, 2,
n)
and let
D = (d..)
be
24
Hermitian of order s.
Then there exists a Hermitian H such
that
ij
A. H A .*
>0
iff
d
j=1
X WT. (i)
k
ij k
k=1
0
, j=1
Proof:
Combine Proposition Z. 4(i) and Proposition 2, 5
Theorem 2. 7. Let
of order
be quasi-commutative matrices
Al' A2'
As
n whose eigenvalues under a natural correspondence are
x(1) x(2) ...,x(s)
k
k
(k = 1, 2,
n)
and let
D = (d..)
be
Then there exists an H > 0 such that
Hermitian of order s.
S
*
d,..A
H A. >0
13
d., X. (i) X (j)
iff
ij k
J
i, j=1
Proof:
k
>0
(k = 1, 2,
n).
i, j=1
Combine Propositions 2. 4(ii) and 2. 5(ii).
Theorem 2. 8. (Schneider)
Let
A, Ck (k = 1, 2,
,
s)
be com-
plex matrices of order n which can be simultaneously triangulated.
Suppose the eigenvalues of
ence are
a.,
y.i(k) ;
i
Hermitian H,
i
7--
A,
1, 2, -
Ck
under the natural correspond-
,n and k = 1, 2,
let
T(H)=AHA
-
CkHCk
k=1
s.
For
25
Then the following are equivalent:
= lail
(i)
(k) 2 > 0
2
lYi
(i = 1, 2,
,n)
.
k=1
There exists an
(ii)
1-1 > 0
such that T(H) > 0,
-1) and apply Proposition
Proof:
Let
2. 5(ii).
Then we have that (i) implies (ii),
D = diag {1, -1, --1,
the submission of [7],
(ii)
In a note added after
Schneider gives a short matrix proof of
(1) by H. Wielandt. We refer the reader to [7) .
It would be nice if we could specialize Proposition 2. 4 for the
part of the Schneider Theorem. However, we have already
(ii)
seen that its quasi-commutativity hypothesis cannot be relaxed to
simultaneous triangulability, even with H > 0,
An alternative
proof of the author's for Proposition 2. 4 can be modified to give this
part of the Schneider result, but it is lengthy and quite computational.
It lacks the elegance of Wielandt's proof.
For Hermitian K,
Tr
(J K
)
< Tr(K)
and
it is known that In J K J < In K. i.e.
v (J K J*) <v (K).
This result was corn-
municated to the author by Carlson. It doesn't seem to appear in the
literature. Thus, we include a proof of the special case which we
shall use.
26
Lemma 2. 9.
n
by
is
s > 1.
sn where
Then
n
by
such tat PJQ = (E0
***
(RTC)) =Q J P = (Er 0
*
J'KJ = (P
-1
where
P
r,
then r < n
P
r = I r(1) 0 n-r
E
.
and
since
Q
Then
and
P),J(QQ
-1
K(Q
Q-1K
P-1(Er0
Since
by
Then there exist non-singular
sn.
be
Tr (J K J ) < Tr(K).
If we denote the rank of J
Proof:
J
Let K be Hermitian of order sn and let J
-1
*
P
Q ),T
-1*
(Er
0.
0)
-1*
)
T
P
-1*
is non-singular, by Sylvester'r theorem we have
that
In J:KJ* = In (Er 0
(Er
where
0)T
L .1= Q-1K(:)- 1*
.== I rEL11E r
I
where
its first
(1)
11
n
is the submatrix of
rows and n
consistin-, of
L
columns
= In(M11+ 0n-r)
where Mill
rows and
Since
M11
[5, p. 203] to get
is the matrix consisting of the first
r
columns of L11
and hence of
is a principal submatrix of L,
we apply
L.
27
(2)
Now
) < Tr(L).
Tr(M11
L=
Q-1K -
implies that
From (1) we have
Tr(L) = Tr(K).
that Tr(JKJ ) =
7(411). Combining these with (2) we have that
or.
Tr(JKJ ) < Tr(K).
We wish to point out that there is no quasi-commutativity or
simultaneous triangulability hypothesis on A 1, A2,
in the
,A
following lemma. In fact, they need not even be square.
Lemma 2. 10.
If
j=1
Hermitian,
Proof:
1313
d.. A .H.A *
Tr(D) < 1,
and
> 0 where
v (D) < 1,
D = (d..)
is
then H is non-singular.
We verify by computation that
.c.-
A -1*
1
d. , A .HA .* = (.A
13
3
1
A2
A2 As)DOH
j=1
=
J (D 0 H) J
where
matrix
Applying Lemma 2. 9, we have that
J
is the
(A1 A2
n
by
A).
sn
28
Tr [ J (D
Since both
ID
0H)J*] < Tr D
and H are Hermitian,
and the eiLenvalues of
H
is Hermitian
are real. Thus by [4, p. 24]
D0H
Tr(D 0 H) = Tr(D)Tr(H) + v (D) v (H).
J > 0,
Tr[J(D H)J]= n.
Tr(H) + v(H)
if
Tr(D)
v(D) = 1
Tr(H)
if
Tr(D) = 1
and
v(D) = 0
v(H)
if
Tr(D) = 0
and
v(D) = 1
Since by hypothesis
J(D
Hence,
n < Tr(D)Tr(H) + v(D)v(H)
We note that if Tr(D) = 0 and v(D) = 0,
since
D
v7e ;:lave that
.
D=0
is Hermitian. Then the condition that
dijA HA.j > 0
j=1
cannot be satisfied and the theorem is true 1r default.
Since H is of order
Tr(H) + v(H) = n
non-singular.
n,
Tr(H) + v(H) < n.
which in turn implies that
6(H) = 0
Thus,
and H
is
29
Let A 1 ' A 2'
Theorem 2. 11.
ces of order
are
(1)
n
whose eigenvalues under a natural correspondence
(2)
matrix of order n,
and let
order
Tr(D) < 1
s
such that
let
,n),
(k = 1, 2,
X(s)
k
'Xk
Xk
be quasi-commutative matri-
s
be a Hermitian
be a Hermitian matrix of
D = (d..)
v(D) < 1.
and
H
If
dijAiHAj > 0,
j=1
then
In H = In {A
Proof:
A
1' A2'
d.. A .HA. > 0
Since
13
3
s
,
D}
.
by hypothesis, by Proposition
j=1
2. 4 we have that
dijXk
(j) /
X.k
0
(k = 1, 2,
n).
j=1
Thus, by Proposition 2. 5 there exists a Hermitian H0 such that
d.. A .H A.*
13
1
0 j
>0
j=1
and
In.H0 = In {A 1' A2'
A
,
D}
.
30
For notational convenience we rename H as H1.
our hypothesis becomes
d..A.H
A. > 0
13 1
1 3
(3)
Then (part of)
.
j=
Let Ht = tH1 + (1-t)H0
where
t E [ 0, 1]
d..A.HtA.3
13
.
Then
nr.
1
j=1
(1-t)H0] Aj
d..A.[tH1
13
1
1, 3=1
d..A.[(1-t)H
13
1
j=1
j=1
d..A.H
A. + (1-t
13
1
1 3
d .A.H
A.
1 0
3
i.3
j=
>0
by (1) and (3) since t and (1-t) are nonnegative and not
simultaneously zero for tE [ 0, ii.
Since
>0
d.,A.H
tA.3
13
j=1
and
D
is Hermitian with Tr(D) < 1
singular by Lemma 2.10. Since
H
and
v(D) < 1,
Ht
is non-
depends continuously on t,
31
where
Ht
tE [ 0, 1]
(which are real since
the eigenvalues of Ht
,
is Hermitian) vary continuously with t. Thus,
singular for t [ 0, 1]
implies that
In Hi = In Ho .
(4)
Since
non-
Ht
H,
H1
we combine (2) and (4) to get
InH = In{A
As,
1' A2'
D}
.
I
The following theory (Lemma 2. 12 and Theorem 2. 13) gives
us a general counterexample which shows that if
v (D) > 1
Tr(D) >
1
or
(or both), then the conclusion of Theorem 2. 11 does not
hold for quasi-commutative A A2' ',As. Thus it completes
the theory in the sense that no less restrictive hypothesis on
will give the InH = In {A1' A2'
Lemma 2. 12:
Tr(D) > 1
AA
1' 2
or
"s
Let
As , D}
be a Hermitian matrix of order s,
D
v(D) > 1,
n> 2,
of order n,
n
which commute in pairs and
such that
d..A.H0 A.> 0
13
j=
If
then there exist complex matrices
a Hermitian matrix H0 of order
and
result.
1
32
H0
Proof:
Suppose
ir(D) > I. Since
a unitary U such that
and
UDU
is Hermitian, there exists
D
diag
112,
are the two positive eigenvalues of
112
hypothesis that Tr(D) > 1
of D.
is singular.
Let
U
ij
and
where
p.
D
guaranteed by the
are the other eigenvalues
P-3,
).
We define the matrices
Al' A2'
and
As
0
H0
by
0
1.3
0
lj
A.
(j
1, 2,
s)
3
0
0
U..
0
u
lj
0
0
.
uli
and
Ho
Then
diag {1, 1, - , 1, 0} where
HO
is of order n.
33
d,. A,H0A,
i.3
1
j=1
u
0
11
13
0
13
j=1
13
u11 ,d..023
.
j=1
0
0
0
.u.
i.3
lj
j=1
S
0
u .d ijTi. lj
0
0
11
i, j=1
s
S
21 13
0
0
13
)-"'
u .d..T.i.
,
i, j=1
21
i, j=1
11
0
0
0
0
FL 1
0
0
1
>0
0
0
II 1
0
0
0
0
}-1,
2
13
.
23
34
s
since
(UDU*
1
=
)lc
_
d..0
u
= p. &
k ki
where
Ske = 0
i,j=1
=1
if k
if k = /.
Thus,
L1
d..A.H
A. >0.
13 i 0 j
j=1
Moreover,
is singular since it has a zero column.
H0
Suppose now that
U
such that
UDU*
v(D) > 1.
Again there exists a unitary
,
= diagih., ti2,
,
P-n} whereand
112
Ill
are now the negative eigenvalues given by the hypothesis that
v(D) > 1.
Letting
we again choose
U = (uij),
uij
0
u,
lj
(j = 1, 2,
0
0
u.2j
0
u
lj
0
0
uij
However, we choose H0 = diag {-1, 1,
which is singular since its
-1, 0}
(step by step) we get
s)
(of order
n)
thcolumn consists of all zeros.
Now by letting the same calculation as in the first
I
(rr(D) > 1)
part
35
A. = diag
3
d..A.H0
13
0
-ill'
1.
j=1
since
and
1.1.2
are negative. Again we have
d..A.H
A.> 0
13
0
1
3
j=1
singular.
with H0
(These two parts of this proof can be combined by letting
Ho = diag { cr, IT,
O}
To show that
where
Cr
commute in pairs, we first
A
A
= sgn
let Enl represent the n-square matrix with a one in its
position and zeros elsewhere. Then Ai = uij I + u2i En
(j = 1, 2,
s).
are both equal to
Thus
Al' A2
By computation we have that A.A.
u11 .0lj
I+
(u21 .0lj + u23 .0li )En1
A
or
2"
As
A.A.
and
s).
(j.,j = 1, 2,
commute in pairs. I
As
Theorem 2. 13. Let D be a Hermitian matrix of order
Tr(D) > 1
(n,l)
v(D) > 1,
of order
then there exist complex matrices
n,
n> 2,
which commute in pairs and
a Hermitian matrix H0 of order n such that
d..
A1 .H
A. >0
13
0
3
j=1
s. If
36
and
InH
A
A
s
,
D}
.12"
By Lemma 2.12 there exist complex matrices
Proof:
of order
order
94 In {..A
n
n
,
Al'
which commute in pairs and a Hermitian matrix H0 of
such that
d..A.H 0 A.*>
1.3
0
3
T.
j=1
and H
is singular. Since Ho is singular,
5(H0) > 0.
How-
ever since
>0
d, ..A..H A .
13
10 j
j=1
and its other hypotheses are satisfied, Proposition 2.4 gives us that
d
.
i3
X
k
X
0
k
(k = 1, 2,
,n).
j=1
i.e.,
In {A 1,A2,
,As,D} = (Tr, v, 6)
InHo
In {Ai,A2,
,As,D}
where
8= 0.
Thus,
.
The following theory for
,
d.,AlHA
where A is a
j=0
complex matrix of order n was developed by the author in the
process of attaininz the previous results of this chapter. This
theory can now be proven as a sequence of corollaries to these
37
previous results.
We call to the reader's attention that to include terms such
as H, HA
Thus,
we must index i and
and A2H
,
D = (d..)
naturally occurs with order
from zero.
j
in the following
s+1
theory.
Theorem 2. 14.
eigenvalues
and let
Let A be a complex matrix of order
let H
X1' X2'
with
n
be Hermitian of order
D be Hermitian of order s+1.
Then the following will
hold:
There exists an H > 0 such that
d.. AlHAJ > 0 iff
dij Xkk > 0 (k = 1, 2,
j=0
n)
.
j=0
There exists a Hermitian H such that
d4k k 0
.*
di.i
A1H AJ > 0
iff
ij
j=0
(3)
If Tr(D) < 1,
(k = 1,2, - ,n).
j=
v(D) < 1,
and
d..AiHAi > 0,
then
j=0
In H = In {A, D} where we define
In {A, D}
to be the ordered
triple of positive negative, and zero values of
d.. X
j=0
j
kk
X
n,
(k = 1, 2,
,n)
.
38
Proof:
Since
,AS
I, A,
AiAi
=
A'
AiAi
A3
(i, j = 0, 1,
s),
commute in pairs and thus are quasi-commutative. We
apply Theorems 2.7, 2. 6, and 2. 11 to obtain parts (1), (2), and (3)
respectively. I
One of the author's results which led to Theorem 2. 14 was
for
+arAr)HAm} where a0' al,
ar are
is a positive integer, m is a nonnegative
Re {(a0I + arA +
complex numbers, r
integer,
A
is a complex matrix of order n,
Hermitian matrix of order
and H is a
The inertia generalization for this
n.
polynomial is interesting. The matrix D of coefficients is
a0
a
2
a
a071
0
1
+
ar
It can be shown by an inductive calculation that this matrix has one
positive eigenvalue, one negative eigenvalue, and
r-1
zero
eigenvalues. (Remember r > 1.)
The coefficient matrices
polynomials are
1
(
0
0 -1
)
and
D
of the Stein and Lyapunov
respectively. An easy
39
computation shows that both matrices have eigenvalues
and
1
-1.
Thus, they both possess corollaries to all three parts of Theorem 2.14.
The Lyapunov and Stein theorems themselves are corollaries
to Theorem 2. 14(1). Parts (2) and (3) for the Lyapunov polynomial
give us the Main Inertia Theorem.
We conclude this chapter with the previously mentioned result
In the in-
of equivalent conditions to simultaneous triangulability.
terest of completeness, the author extends a theorem of Drazin,
Dungey, and Gruenberg [2, p. 222] to six equivalent conditions.
One of these doesn't appear in the literature. However, it did appear
naturally in the development of this thesis:
If
Al' A 2'
s
satisfy the condition of Lemma 2.3, we say that they are
E - simultaneously triangulable. This is condition (iv) below.
Theorem 2. 15. Let
n.
be complex matrices of order
Al' A2'
Then the following statements are equivalent.
For every polynomial p(x1, x2,
,xs) in the non-commutative)
variables x 1 x2
xs each of the matrices
(i)
p(Al, A2,
,A s)(A.A.
13 - A.A.)
3
I
There is a unitary matrix
is nilpotent
U
(i, j = 1, 2,
s).
such that UAU is tri3
angular
(j = 1, 2,
,
s).
There is a non-singular P such that
P'AP
3
is triangular
40
(j = 1, 2,
s).
E > 0,
Given
S-1A.S = A. + T.
J
3
there exists a non-singular
where
strictly upper triangular with
for
I tLe(j) I < E
There is an ordering of the eigenvalues
,n; j = 1, 2,
(k = 1, 2,
rational function R(Ai,
(2)
R(XkXk
(vi)
'
(s))
s)
,n;
i, 1 = 1, 2,
,n; j = 1, 2,
(k = 1, 2,
,As)
(k = 1, 2,
s)
polynomial function P(A1 'A2'
(1)
(2)
P(Xk ,Xk '
(s) )
(k = 1, 2,
X
(i)
of each A.
such that the eigenvalues of any
A1'A2',As are
of
,
s).
There is an ordering of the eigenvalues
Proof:
is
s;i>i.
= 1, 2,
(1)
such that
is diagonal and T.3 = (t.11 (i))
A.
3
3
S
X
(i)
of each A.
such that the eigenvalues of any
,A)
of
A1'A2'
As
are
,n).
The Drazin-Dungey-Gruenberg theorem gives us the equi-
valence of (i), (ii), and (v). Since a unitary matrix is non-singular,
(ii) implies (iii). Lemma 2.3 is the statement that (iii) implies (iv).
The proof that (iv) implies (v) is the same as that of (ii) implies (v)
since in both cases the given triangulation exhibits the eigenvalues
of the matrices concerned on the main diagonal. Clearly, (vi) is a
special case of (v). The Drazin-Dungey-Gruenberg proof of
" (v)
(i)"
uses only condition (vi) as a hypothesis. I
41
CHAPTER 3: THE THEORY FOR GENERAL
MATRIC FUNCTIONS
The theory developed in the first part of this chapter follows
a pattern similar to the theory of Chapter 2. Again the results of
Lyapunov, Stein, Sylvester, and the Main Inertia Theorem are
generalized as well as those of [3]
.
Letting A be a complex matrix of order n;
Hermitian matrix of order n;
defined on A;
A
and
fl'
,f
H
be a
be matric functions
gm be matric functions defined on
g 1,
(see the following paragraph), we develop a body of theory for
fi(A)H g1(A)
i=1
which plays the role of
d..A.HA.
13
1
*
j=1
in the theory of the previous chapter. The author first developed
these results for
pi (A) H qi (A )
i=1
where
pi, -
'n
and
qi,
,q
are polynomials.
However,
42
the theory for
A H q. (A)
Pi().
i=1
is better discussed in terms of
d
ij
Ai H Aj
j=0
where we multiply out our polynomials and combine like terms.
We have already discussed the theory for
d. . AiH Aj
*
13
j=0
in Chapter 2, deriving our results as corollaries to the results for
13
1
3
1, j=1
where
Al' A2
A
s
are quasi-commutative.
Furthermore,
we find a characterization of the "Hermitian-preserving" (see p.45
for a definition) hypothesis for
( Ap.
)H qi(A )
i=1
to be that the matrix of coefficients
(d..)
13
in the
43
d,.A1HAJ
j=0
form is Hermitian.
In this section most of our results involve "matric functions.
We use a standard definition. See [4, p. 73] . Given a complex
matrix A of order
f(A)
kth
n
and a complex-valued function of f(z),
is defined iff f(k)(X ) is defined
derivative) for each eigenvalue
(where
(k)
denotes the
of A where
X
is the elementary divisor of highest degree involving
z-X
(z-X )k
.
The reader may wish to consider matric functions to be those
which are infinitely differentiable. This change won't affect our
development of the following theory.
Proposition 3.1.
eigenvalues
let
gl,
Let A be a complex matrix of order
Xi, X2,
Xn;
let
gm be defined on A ; and let
H
be a Hermitian
matrix of order n such that
gi
)>0
.
Then
ik ik
If (X )g.(T. )
i=1
with
be defined on A;
fl'
fi(A)H .(
n
0
(k = 1,2,
,n).
44
Proof: Our proof is by contraposition. We assume that A has an
eigenvalue
such that
X
fi(X)gi(T.) = 0.
i=1
Since
X
is an eigenvalue of A,
X
and there exists a (row) vector v
-*
*
NT
=Xv.
= 1, 2,
Then vfi(A) = fi(X)v
m).
See [4, p. 73]
,
is an eigenvalue of A
0
such that vA = Xv and
and
-
g.(AA hr = gi())v
Now
-(i
...t.
vfi(A)] Fir gi(A )v ]
f(A) H gi(A ),1 v
i=1
*
f.(X)v] H [g.(X)v ]
f(x)g(x)1 vH v*
i=1
0.
By [4, p. 69]
fi(A) H gi(A )
i=1
is not positive definite. This is a negation of our hypothesis. I
Definition 3.2.
A function
F(A, A ,H) =
fi(A) H gi(A
i=1
)
is said
45
to be Hermitian-preserving iff the image is Hermitian for all com-
plex A and Hermitian H of order
n
for which the image is
defined.
Definition 3.3.
eigenvalues
Let A be a complex matrix of order
let f1,
Xn;
X1' X2'
n
with
be defined on A;
be defined on A. We define
and let
In {A ,
fl,
where
it
fm, gi,
gm}
to be the ordered triple
is the number of positive values,
negative values, and
v
(Tr
,v, 5)
is the number of
is the number of zero values of
5
f.(Xk
i
i=1
ik
) g.(T. )
(k = 1, 2,
,n)
.
We remark that
f.(X
i=1
g(X)
kik
,n) if this definition is to make sense.
must be real (k = 1, 2,
Let
fi(A) H gi(A*)
i=1
be Hermitian-preserving.
n
for which
Let A and H
be matrices of order
46
H gi(A
)
i=1
is defined. Then for each
,n,
k = 1, 2,
f.(Xk
)
and
are defined; and taking A = Xk (and H = 1),
fi(Xk) gi(
i=1
is defined and Hermitian (i. e.
Proposition 3. 4.
eigenvalues
,
real).
Let A be a complex matrix of order
X1, X2,
n
with
If
Xn.
fi(A)H gi(A )
1=1
is Hermitian-preserving and if
ik ik
) g (T.
i=1
)
0
(k = 1, 2,
then there exists a Hermitain matrix H0 of order
n),
n
such that
fi(A) Ho gi(A) > 0
(i)
i=1
and
(ii)
In H
In {A, fi,
,fm,gl,
gm}
.
47
Proof:
E>0
Given
a non-singular
S
such that
e (xklk
s--4As
where
Uk
there exists
and our complex matrix A,
E Uk)
is a square matrix of the order of
Ik
with ones on the
first superdiagonal and zeros elsewhere. Renaming S-1AS =
* _1*
we have that A =SAS
.
Then f() -= S-lf(A)S
KO= eg(A*)S-1*. See [4, p. 73]
.
and
We let
111
sk = sgn
fi(X
)
1, 2, '
(k
,n)
.
i=1
Since
fi(A)H gi(A*)
i=1
is Hermitian-preserving,
fi(xk)
i=1
is real (k = 1, 2, -
,n).
Thus,
fm, gl,
In {A, fi,
(See the remark following Definition 3. 3.)
Hermitian where we define
gm}
1-1
A
is well-defined and H is
by
diags
1' 2'
s.}
n
.
48
By this definition,
Ini
= In {A,f
,fm,gl,
,
gm}
.
Now
fi (a)
11
i=1
111
= diag {
f.
inn
f. X
i=1
i=1
gi (T. )1)-1- M.
E
To verify this statement we compute
fi (X) I-I gi(A.'3*)
i=1
elementwise. This is a straightforward but long calculation. Written
out elementwise, we see that every term of (2) above not in diag
has a factor
A similar calculation was written out in
Proposition 2. 5.
Since
f. (A) H .(
i=1
is Hermitian-preserving by hypothesis,
}
49
111
f(X) fi
*:)
1=1
is Hermitian. Since
f.(X )g.(K. )
(k = 1, 2,
0,
rx)
the expression (2) is positive definite for a sufficiently small choice
e.
of
fi
Thus
gi(A.j*)
)
*
*
1
=
)f.(A S] H [S g.(A )S
]
>0.
i=1
1=1
Since
-
is non-singular and S-1
S
=
-1
S*
we apply Sylvester's
theorem to get
S
-
Alf.( )S1
*
n *
H [S gi(A )
m
,
..T.
..].
*
fi(A)Hogi(A ) > 0
] )S =
1.
1=1
where
Ho = SHS .
Once again the Sylvester theorem gives us that InH 0 = InH.
Combining this with (1) we have that
InHo = In {A, f1,'
fm, gi, -
gm}
.
I
Theorem 3.5. Let A be a complex matrix of order
eigenvalues
X1' X2'
Xn
and let
n
with
50
f.( )H gi(A
i=1
be Hermitian-preserving. Then there exists a Hermitian matrix H
of order
n
such that
fi(A)H gi A ) > 0 iff
i=1
f (N.k) gi (K.k)
0
(k = 1, 2,
,
n).
i=1
Combine Proposition 3. 1 and Proposition 3. 4. I
Proof:
Theorem 3. 6. Let A be a complex matrix of order n with
eigenvalues
Xi_ , X2 ,
and let
X.n
fi(A) H gi(A )
i=1
be Hermitian-preserving. Then there exists an H> 0 such that
m
m
)H
i
gi
(A
*
)>0
iff
f.
i
i=1
Proof:
(Xk)> 0 (k =1, 2, -
i=1
(Only if)
Let
the (row) eigenvector
be an eigenvalue of A corresponding to
X
v.
i. e.
vA = Xv.
,
f
(1)
A.( )H gi(A
4)] v*
[ v fi(X )] H [g.() v}
i=1
fi(X )gi(T)1 H v
=[
1=1
*
Then A v* = X v
111
i=1
,n).
*
and
51
Now H > 0 implies that vH v> 0 and
fi(A)H gi(A ) > 0
i=1
implies that
*
Af.( )Hgi(A )j v > 0.
i=1
Thus, from equation (1) we have that
Ifi(X )g1(X)
i=1
is positive for all eigenvalues
X
of
fi(xk)gi()Tk) > 0
A.
i. e.
(k = 1, 2,
n).
1=1
(If)
Assuming that
ik ik ) > 0
f.(X ) g
1=1
(k
1, 2,
,n)
and following the path of the proof of Proposition 3. 4, we have that
sk = sgn
fi(Xk)gi(Tic)] = 1
(k = 1, 2,
,n).
1=1
Thus,
a 7- In
and Ho = S S* > 0.
Ho we have that
For this positive definite
52
fi(A)H0 gi(A ) > 0 .
A rewording of the condition that
fl(xk)gi(k)
is that In {A, f 1,
gi ,
°
(k = 1, 2, "
n)
gm} = (n, 0, 0). Thus Theorem. 3.6
naturally generalizes the Lyapunov and the Stein theorems as well
as a specialization of the Schneider theorem, whereas Theorem 3. 5
is a natural generalization of the first part of the Main Inertia
Theorem.
We pause here to remark that it is possible to have H > 0
and
k ik
f. (X ) g.(X ) > 0
i=1
(k = 1, 2,
,n)
where
fi(A)H gi(A )
i= 1
is Hermitian-preserving and yet have
Tri
fi(A)H gi(A )
i=1
1
indefinite and non-singular. Let A = (3 0)
1
and H
Then the
53
are
eigenvalues of AH + HA
5
-1,
and
which implies that
is indefinite and non-singular.
AH + HA
We have observed that once again the inertial formulation is
the same as the "iff" solution. Theorem 3. 6 can be expressed in
either form. However, the inertial problem for Hermitian H and
general f1,
,fm, gl,
fm, gl,
fi,
, gm
gm
is unsolved. For polynomial
the problem is solved by Theorem 2. 14(3).
We now give another partial solution. This one is used as a lemma
for some of the applications of the next section.
Let A and H
Definition 3. 7.
n.
A function F(A, A ,H)
be complex matrices of order
is said to be accessible iff it can be
expressed as a linear combination of functions each of which has
H
as an initial or terminal factor.
Lemma 3. 8.
H
Let A be a complex matrix of order
be a Hermitian matrix of order n.
and let
n
If
fi(A) H gi(A
i=1
is accessible and positive definite, then H is non-singular.
Proof:
singular.
Our proof is by contraposition. We assume that H is
Then
0
is an eigenvalue of H and there exists a
corresponding (column) eigenvector
x 4O
such that
Hx = Ox = 0.
54
*
Since H is Hermitian,
*
*
xH=xH=0
.
Now using our
accessibility hypothesis and reindexin.g, we have that
x[
fi(A)H gi(A flx = x
i=1
H gi(A ) +
fi(A)111 x
i=1
[x H] gi(A )x +x
i=1
i:=1 +1
gi(A ) x+
r^.
fi(A)[Hx]
fi(A)
i4+1
i=1
= 0.
Hence,
by [4, p, 69],
.(
Af.(A)H
is not positive definite.
)
i=1
This is a negation of our hypothesis.
Theorem 3.9.
H
Let A be a complex matrix of order
be a Hermitian matrix of order n.
n
and let
If
fi (A)H gi(A*)
i=1
is positive definite, Hermitian-preserving and accessible, then
In H = In {A, fi,
Proof:
Let
X1' X2'
Xn
fm, gl,
gm}
.
be the eigenvalues of A.
Since
55
rn
Ifi(A)H gi(A ) > 0
i=1
by hypothesis, we have that
111
fi(xidgiOc) 4 ° (k = 1, 2,
by Proposition 3. 1.
,n)
We now apply Proposition 3. 4 which gives us
that there exists a Hermitian matrix H0 such that
fi (A)H0g i
(A*)
>0
i=1
and
In H0 = In {A, f 1,
fm, gi,
gm}
For notational convenience we rename H as H1.
we then have that
Ii(A)Hi gi(A ) > 0
(3)
i=1
Let Ht = tH1 + (1-t)H0 where
t E [ 0, 1
.
By hypothesis
56
AIf.(
)H g.(A*)
t
fi(A)[tHi + (1-t)H0] gi(A*)
1=1
111
fi(A)[tHi]l gi(A) +
1=1
fi(A)[(1-t)H
gi(A*)
1=1
111
Af.( )H g(A ) > 0
f.1(A)1-1 1 gi(A ) + (1-t
i=1
1=1
by (1) and (3) since both t
(1-t)
and
are nonnegative and they
are not simultaneously zero for any te [0, 1] .
Since
fi(A)Ht gi (Am)
1=1
is positive definite and accessible,
3. 8.
Since H
(which are real since Ht is Hermitian)
vary continuously with t.
In H =
is non-singular by Lemma
depends continuously on t where t E [0,1]
the eigenvalues of Ht
InH1 = InH0*
Ht
Since
fi,
Thus,
°
nonsingular implies that
combining this with (2) gives us
,
H1
Ht
g
,
57
The following theorem gives a characterization of our
Hermitian-preserving condition for polynomials
Condition (iii) of the following theorem
pi,
is simply that the matrix coefficients
(c1..)
be Hermitian in the
.*
d..
j=0
form of
pi(A)H qi(A ).
1=1
This result helped motivate some of the results of Chapter 2.
If
p(X ) = aX n + an_ X n-1
alX + a 0'
+
we shall
define
600 = anXn+ a n- 1X n+1+
1
X +ao
Theorem 3.10 is more simply stated if we let
maximum of the degrees of pi,
number of Pils
, Pm, qi,
be the
s
qm and m, the
(and thus ofs.' ). No generality is lost since
we can fill in with zero polynomials and/or zero coefficients of the
higher terms of the given polynomials.
Theorem 3.10. Let A be a complex matrix of order
H
be a Hermitian matrix of order n.
Let
n
and let
58
X
Pi.()
=a
.X
s
(i = 1, 2,
(i)
+
si
s).
a iiX + aoi and
s
qi(X.) = b .X +
biiX + boi
si
Then the following conditions are equivalent:
The function H
A H q.(A ) is Hermitian-preserving.
Pi().
1=1
pick) (x )qiCi)(x )
tiick)(x ) 1;0) (x )
n-1).
i=1
1=1
a
.
b..
i=1
Proof:
(k, j = 0, 1,
b
=
. a..
(k,j = 0,1, -
,n-1).
1=1
Since A and H are n-square matrices with H
Hermitian, by [7] a matrix representation of the operator
H
qi(A*)
is
p.(A) 0 qi(A) = pi(A) 0 qi(A)
.
Thus a
matrix representation of
pi(A) H qi(A*)
i=1
is
0 q.(.
A-.)Ipi(A)
1=1
pi(A)H qi(A*) to be Hermitian for all A and H,
Now for
1=1
pi(A) H qi(A*)
1=1
must be equal to
59
A H qi(A
Pi().
(A) H ci(A44)
=
)
i=1
.
1=1
This condition implies that
pi(A) tv.
(1)
qi(A)
%(A) =
pi (-A-)
1=1
We now specialize 4 to an arbitrary Jordan block of ordern:
0
1
0
OX
1
0
0
0
1
0
0
0
X
X
A=
We multiply out our transformation matrices of (1) and equate elementwise. This gives us condition (ii).
Assuming that condition (ii) holds, in particular it holds
when
Since
X = 0.
pi(0) = aoi,
p
p.(X ) =a .X 5+
si
(0) =
a1i"
).(Xsi=bs+--+b11X +b 01,
q.(0) = bo., q.' (0)
Thus
bli,
+a .X +a 01.'
(k)(0) = k! aki,
i
we have that
pi(s Cs! asi.
we have that
qi(j)(0) = j!, bii,
qi(s)(0) = s ! b
Since
60
k! a.k )(j! b..)
pfr)(0)q(j)(0)
1=1
and
,(k)
( )p
q
j)
(0) =
(k, j = 0, 1,
k! i-s-)ki)(j!
n-1) .
i=1
Condition (ii) for
ab
ki
implies that
X=0
(k,j = 0,1,-
. a..31
.
i=1
,n-1).
i=1
We now show that (iii) implies (i). Since
pi(A) = asi.As+q
*
+akiAk+
+aoiI
-
and
*s
) = bsiA. + - +b..A.'
+i(A
-
0
31
.1,
in
pi(A)Hqi(A*)
i=1
is
/
a b...
ki
i=1
We also have that
1-3-ki:ji
i=:1
is the coefficient of
Alc1-1A*i
in
the coefficient of A- HA
*3
61
pi (A)H
i=1
Thus, (iii) implies that
pi(A)H q A )
(A) H
i=1
.( Ap.
)
.
i=1
i.e.
pi(A)H qi(A*)
i=1
is Hermitian for all A and
H.
Thus, the function
pi(A)H qi(A*)
i=1
is Hermitian-preserving. I
62
CHAPTER 4. SOME ANALYTIC GEOMETRY APPLICATIONS
It is always interesting to tie together different branches of
mathematics. In this section we combine some of the matric results
of this thesis with some basic analytic geometry.
The author developed many of these results in [3]
,
directly
proving them from the Main Inertia Theorem. Nevertheless, these
results naturally follow the results of Chapter 3 as they are corol-
laries to Theorems 3.5, 3. 6, and 3. 9.
We begin this development with two lemmas which are them-
selves corollaries to the above-mentioned theorems. Many of the
applications of this section are specializations of one or the other
of these lemmas.
In the following we let
re {f(X)}
-1-{f(X) + f(3-0 J
.
im{f(X )}
-liff(X)-fi-in
Then, as before,
InIA, im fl and
In {A, re f} are defined to be the ordered triples
It
is the number of positive values,
values, and
re {f(X k)}
6
v
(k
1, 2,
,
(n, v, 6)
where
the number of negative
the number of zero values of
respectively
and
im {f (X k)}
n)) where Xi, X2,
and
Xn
are the eigenvalues of the matrix A.
Since Lemmas 4.1 and 4. 2 are structurally the same, we
choose to combine their proofs.
63
Let A be a complex matrix of order n and let
Lemma 4. 1.
f be any matric function for which f(A)
and
are defined.
f(A )
Then
There exists an H > 0 such that Im{f(A)H} > 0 iff
im {f(X k)} > 0
(k = 1, 2,
n)
where
Xn are the
Xi, X.2,
eigenvalues of A.
There exists a Hermitian H such that Im{f(A)H} > 0 iff
A has no eigenvalues
If
H
such that f(X)
is real.
is Hermitian and Im {f(A)H} > 0,
InH = In {A, imf}
then
.
Let A be a complex matrix of order
Lemma 4. 2.
f
X
be any matric function for which f(A)
and
f(A
n
and let
are defined.
)
Then
There exists an H > 0 such that Re {f(A)H} > 0 iff
re {f(X k)} > 0
(k = 1, 2,
,n)
where
X
Xl' X2'
n
are the
eigenvalues of A.
There exists a Hermitian H such that Re {f(A)H} > 0 ill
A has no eigenvalues
If
H
X
such that f(X)
is Hermitian and
In H = In {A, re f} .
is (pure) imaginary.
Re {f(A)H} > 0,
then
64
We first observe that Im {f(A)H } and Re{f(A)H} are
Proof:
Hermitian-preserving. Thus, Theorem 3. 6 gives us (i) and Theorem
3.5 gives us (ii). Since
Im {f(A)H}
and
Re {f(A)H}
are accessi-
ble as well, Theorem 3. 9 gives us (iii).
At times in the following development it will be convenient to
r associated with a particular application.
talk about the "curve"
We shall define the "curve"
r to be the point set from which
eigenvalues are excluded in the (ii) form of our results.
would be the point set
Lemma 4. 1,
If
lz: im {f(z)} = 0}
e. g. , in
.
r separates the complex plane into two disjoint parts,
applications of (iii) will say that
In H
will be equal to the ordered
triple of nonnegative integers formed by the numbers of eigenvalues
of A which lie outside, inside, and on F.
2"
of A for which
the eigenvalues
X
(k = 1, 2,
is positive, negative, or zero.
n)
Xn
viz., the number of
If
k
im {f(X)}
r is a line,
circle, or hyperbola, this gives us a convenient language for discussing the applications.
Each of the following applications can be expressed in the
form of the three above results. We choose to express them in
the second form.
Application 1:
If
A
is a complex matrix of order n,
then
there exists a Hermitian H such that Re{e-i°AH-e-iep.H} > 0
65
iff
A has no eigenvalues
where
- 00< r <00,
Proof:
We let f(A)
on the line z = p. + r ei(0 +
z
0 < 0 < 2Tr,
and
pt.
)
is complex.
e-i°(A-p,I)) where 0 < 8 + 27T and
is
complexjand we apply Lemma 4. 2. Now A has no eigenvalues
such that
If
z
re {f(z)} = re {e-i0(z-1.1.)} = 0.
+ r ei(0+
z
re {e(z-p.)}
-18
-2- )
where
rele'40 (r e i0
- co< r <00,
then
71.
2)}
Tr
= re{r e
2}
Conversely, if re {e-ie(z-p.)} = 0,
-oo < r
<00
Thus,
= re {ir} = 0.
then e-ie(z-ii.) = r i where
z-p. = r ie10 = re
)
or
z = p.+rei(0-1-I)
Application 1 was motivated by an arbitrary line in the com-
plex plane. As discussed in [3] , this result may be specialized for
a line parallel to either axis or for one intersecting both axes.
i. e.
,
given any equation representing a line in the complex plane,
we have a theorem corresponding to it. Similarly, Application 3
was motivated by an arbitrary circle.
However, Application 5 was not motivated by a hyperbola
but rather by the function f(A)
aA2 + 13A + y I.
The remaining
applications were also motivated by special choices of
f(A).
It might be conjectured that for every curve which splits the
complex plane into two parts there exists a result of the type given
in the applications. However, this is not the case. Our type of
theorem must be modifed for as simple a curve as the parabola.
This is discussed in [3].
We remark that if we specialize Lemma 4. 2 to f(A) = A,
then (i) is the Lyapunov theorem whereas (ii) and (iii) give both
parts of the Main Inertia Theorem. We could specialize Application
1 to the same results by letting 0 = 0 and
1.1.
= 0.
Because of
the significance of these results we again give them a listing in our
application form.
Application 2.
If
is a complex matrix of order n,
A
then
there exists a Hermitian H such that Re{AH}> 0 iff A has
no eigenvalues on the imaginary axis.
Application 3.. If
(real), and
that H -
er
1
on the circle
is a complex matrix of order n,
A
r>0
is complex, then there exists a Hermitian H such
*
(A -a.I)H(A -cr I) > 0
k-
iff
A has no eigenvalues
r.
I
Proof: We apply Theorem 3.5 with f1(A) =I, f2(A) =
) = I, and
g1
gz(A
complex. Then
(A
z
*
)
A ,-0" I,
where r > 0
and 0" is
I),
67
2
1 fi(z)gi(;)
1
= 1 ---.2-(z- o-)(z-a.)
r
i=1
01
iff
--2-
lz-o-12
iff iz-o-,=r
1
I
I
r
If we let f1(A) = I,
f2(A) = A,
gi(A ) = I,
and
g2(A )=A
then Theorem 3. 6 becomes the Stein Theorem whereas Theorems
3. 5 and 3. 9 become the Ostrowski-Schneider analogue of the Stein
Theorem. (See [3, p.
15]
letting r = 1 and
=0
a-
.
We could also obtain these results by
)
in Application 3. Again we list this
result because of its significance.
Application 4.
If
A
is a complex matrix of order n,
then
*-
there exists a Hermitian H such that H - AHA > 0 iff A has
no eigenvalues on the unit circle
Iz
i. e. , no eigenvalues
z
such that
= 1.
Application 5.
If
A
is a complex matrix of order n and
and y are complex constants where
y = e +fi;
o,
p = c+di,
and
then there exists a Hermitian H such that
Im {(aA2 + f3A+yI)H } > 0
on the hyperbola
Proof:
a = a+bi
f3
iff A has no eigenvalues
z
x+iy
bx2+ 2axy - by2 + dx+ cy + f = 0.
We use Lemma 4.1 withf(A) = aA2+13A + yI.
Now
im {f(z)} = im{az2 +13z + y} = im{(a+bi)(x+iy)2+(c+di)(x+iy)+(e+fi)}
=
bx2 + 2axy -by2 + dx + cy + f
=
0.
*
68
Since
(2a)2 - 4b(-b) = 4a2 + 4b2 > 0,
the locus is a hyperbola (or
two intersecting lines in the degenerate case. )
The author conjectures but has been unable to prove that if
a, p, y, and
that
are complex, then there exists a Hermitian H
6
Im{(aA3
+ 13A2
such
iff A has no eigenvalues on
+ yA + 6I)H } > 0
a locus which is the union of a hyperbola and a straight line (three
lines in the degenerate case). This conjecture can be shown true in
special cases. Our next application is one of these.
Application 6. If A
is a complex matrix of order
a, b, c,d
and
n
are real constants, then there exists a Hermitian H such that
Im{ (aA3 + bA2 + cA + dI)H} > 0
on the x-axis and the hyperbola
iff
A
has no eigenvalues
3ax2-ay2 + 2bx + c = 0.
Proof: We apply Lemma 4. 1 to f(A) = aA3
im{f(z)} = im{az3+bz2+cz+d}
We observe that
bA2 + cA + dl.
a(3x2y-y3)+ 2bxy+cy
=
y(3ax2 - ay2 + 2bx + c) =
3ax2 -ay2 + 2bx+ c = 0
a=c=1
+
=
bola (or a degenerate case thereof).
Letting
z = x+iy
0
Then
.
is the equation of a hyper-
I
in Application 6 we get the following
inertial picture where A has
Tr
eigenvalues in the unshaded
region, v eigenvalues in the shaded region, and
5=0
eigenvalues
69
,o
20/
on the hyperbola and x-axis.
'/Pc
///
/,//////
///61/1///
'///////////',/,///,
/////,/,/4
/ /// /////,// /////,/,.///////4//,
///,/////////////////// /////////////////////
Figure 1.
Application 7.
If
is a complex matrix of order n and
A
a positive integer, then there exists a Hermitian H
Im{ArH} > 0
iff
A has no eigenvalues
(through the origin)
Proof:
Let f(A) = Ar
eigenvalues
arg z
arg z =
2Trk
z
21rk
(k = 0,1,
z
on the
such that
r
lines
-,r-1).
and apply Lemma 4. 1. Then A has no
such that
zn
is real, i. e. , such that
(k = 0,1,- -,r-1).
We observe that Application 7 is much nicer in its
form than its
r is
Re {- }
form since the
nth
root (rth
Im{- }
root in our
result) of an arbitrary real number is more simply expressed than the
nth
root of an arbitrary imaginary number.
The type (iii) analogue of this result is particularly interesting.
We include pictures of the special cases where r = 2, 3, and 4.
70
When
our result is a degenerate case of the hyperbola of
n=2
Application 6, viz, the coordinate axes. As before,
eigenvalues in the unshaded region,
region, and
6=0
v
previously mentioned cases:
and
im{z
=
Tr
r
separating
r for each of the
im{z2} = 2xy;
4x3 y 4xy3 =4xy(x+y)(x-y)
has
eigenvalues in the shaded
eigenvalues on the "curve"
them. The following calculations give us
A
im {z3} =3x2y-y3;
where, as usual,
z = x+iy.
---
7
-
17--
7--
Im {A2H} > 0
Im {A4H}> 0
Figure 2.
Whenever elementary functions are considered, some of the
first to come to mind are the exponential, the trigonometric, and the
hyperbolic functions. We now investigate the .nertial loci which some
of these functions yield.
71
Application 8. If
then there
is a complex matrix of order n,
A
exists a Hermitian H such that Im
eigenvalues on the lines
where
y = kir
>0
iff
A
has no
is an integer.
k
Proof: We apply Lemma 4. 1 to f(A) = eA. Since
ez exeiy ex
(cos y + i sin y), im fez) = exsinY. Thus,
im{ez} =0
y = kir
implies that sin y = 0 which in turn implies that
where
k
Moreover,
is an integer. I
im{ez} > 0 implies that sin y > 0.
This
gives us the following inertial graph:
5ir
X
4
Re{eAH} > 0
Figure 3.
Application 9.
If
A
is a complex matrix of order n,
then
there exists a Hermitian H such that Irri{Oin A)H} > 0 iff
has no eigenvalues
z = x+iy
on the lines
y
0
or
x = irk +
A
Tr
72
where k is an integer.
Proof:
We again use Lemma 4. 1, applying it to f(A) = sin A.
sin z = sin (x+iy) = sin x cosh y + i cos x sinh y,
Since
irn {sin z} = cos x sinh y.
im{sin
If
which in turn implies that y = 0
= 0,
or
Tr
x=
then cos x sinh y=0
+ kTr where
k
is an
integer.
Since
implies that cos x sinh y > 0,
im{sin. z} > 0
our
inertial graph is as follows:
1
5Tr
2
-
Tr
41
Tr
3Tr
2
2
it
111'
111
11
Re{(sin A)H} > 0
Figure 4.
Since
sin z = sin x cosh y + i cos x sinh y,
cos z = cos x cosh y -i sin x sinh y,
our inertial regions corre-
sponding (under the usual form of the application) to
and
and
Irn {(sinA)H}
Re {(cos A)H} are infinite half-strips (See Figure 4) whereas
our inertial regions corresponding to
Re {(sin A)H} and
73
Im{(cos A)H} are infinite strips (as are those corresponding to
Re{eAH}
and
Im{eAFI} ).
fact that cosh r
(for real r).
The reason for this difference is the
is never zero whereas
sinh r
0
iff
r
0
Moreover, cosh r is always positive; sinh r > 0 iff
r > 0.
This phenomenon is deterministic in our hyperbolic applica-
tions as well. Since
cosh z
sinh x cos y + i cosh x sin y and
sinh z
cosh x cos y + i sinh x sin y,
sponding to
Re {(sinh A)H}
and
the inertial regions corre-
Im {(cosh A)H} are infinite
half-strips whereas the inertial regions corresponding to
Re {(cosh A)H} and Im {(sinh A)H} are infinite strips.
Many more applications could be considered. It is apparent
that these results could be generated ad infinitum. The author has
given a sample of those applications which appeal to him as
interesting, curious, or "nice"
others.
.
The reader may wish to investigate
74
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Hill, Richard D. Generalizations of the Ostrowski-Schneider
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State University, 1965. 31 numb. leaves.
Marcus, Marvin and Henryk Minc. A survey of matrix theory
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Introduction to linear algebra. New York,
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Ostrowski, Alexander, and Hans Schneider. Some theorems on
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Analysis and Applications 4:72-84. 1962.
Schneider, Hans. Positive operators and an inertia theorem.
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Stein, P. Some general theorems on iterants. Journal of
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1952.
Taus sky, Olga. A remark on a theorem of Lyapunov. Journal
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