AN ABSTRACT OF THE THESIS OF
for the
CHARLES GORDON LINDBERG
(Name)
Mathematics
(Major)
in
Master of Science
(Degree)
August 31, 1967
presented on
(Date)
Title A CONE ASSOCIATED WITH THE LYAPUNOV MAPPING
Abstract approved:
Redacted for Privacy
(C. S
In this
+
In
of the cone of
under this mapping. In Section
matrix
`
hermitian matrix
A.
P
is a hermitian
particular, for special positive stable
characterize the image
nX n
PA
is a positive stable matrix and
A
matrix.
Ballantine)
paper we investigate the Lyapunov mapping
P -- AP
where
.
H
V
A
we
positive definite matrices
we give five conditions on an
and a general
nXn
that are equivalent to the condition:
for some positive definite (hermitian)
P.
positive stable
H
=
AP
+
PA
A Cone
Associated with the Lyapunov Mapping
by
Charles Gordon Lindberg
A THESIS
submitted to
Oregon State University
partial fulfillment of
the requirements for the
in
degree of
Master
of
Science
June 1968
APPROVED:
Redacted for Privacy
Associate Professor
of Mathematics
In Charge of Major
Redacted for Privacy
Chairman of Department
of
Mathematics
Redacted for Privacy
Dean of Graduate School
Date thesis is presented
August 31, 1967
Typed by Carol Baker for
Charles Gordon Lindberg
ACKNOWLEDGMENT
This author is indeed indebted to Dr. C.S. Ballantine for his
guidance and encouragement during the work on this thesis.
TABLE OF CONTENTS
Page
I.
INTRODUCTION
II.
SOME PROPERTIES OF THE MAPPING
III.
[A
IV.
[A]sfP
V.
]a)
1
FOR DIAGONAL
FORA GENERAL
CONDITIONS ON
HE[A]P
BIBLIOGRAPHY
H
AND
P
_ AP+ PA''`
A
6
15
2X2
A
A
25
EQUIVALENT TO
32
42
A CONE ASSOCIATED WITH THE LYAPUNOV MAPPING
A
large amount
I.
INTRODUCTION
of
research in matrix theory in the past few
years has been concerned with the matrix equation
Y
=
Lyapunov in his 1892 monograph
of Motion 1/
[
8]
AX
+
XA*
(1.1)
.
The General Problem of Stability
established that the equation
(1. 1) could be used to
obtain information pertaining to the location of the characteristic
values
a
of
variety
A.
of
More recently, interest in equation (1.
directions.
A
has taken
survey of results linked with this matrix
equation can be found in Givens' report
Elementary Divisors and
Some Properties of the Lyapunov Mapping
70]
1)
X
-
AX
+ XA
[
5, p. 62-
.
One
direction
of study was
suggested by Olga Taus sky in the
Bulletin of the American Mathematical Society [ 15, p. 711] when
she queried:
Let A be an nXn matrix with complex elements
and characteristic roots with negative real parts. It is
known (theorem of Lyapunov) that such matrices are
1/
The translation of the title of Lyapunov's monograph used here is
taken from Givens [5, p.
3]
.
2
characterized by the fact that
a positive definite G
exists with AG + GA* negative definite. What is
the range of AG + GA* if G runs .4through all
positive definite nXn matrices? A is the
complex conjugate and transposed matrix.
This thesis addresses itself to this question. After considering
some of the properties of the matrix equation (1.
mapping, we characterize the set
as
V
P
=
AP
and the related
PA* for some special
A,
ranges over the set of positive definite matrices. In Section
and a positive stable
A
of
+
(Theorem 6) we give five conditions on an nXn
H
H
AP
1)
+
PA''`
that are equivalent to the condition:
A
for some positive definite
brief background
of
hermitian matrix
equation (1.
1)
P.
is given now for the sake
completeness. As noted, Lyapunov is credited with the use
the equation (1.
1)
of
for the location of characteristic values. What he
did was to establish a connection between the location of the charac-
teristic values
of a
general
matrix and the signature
nXn
quadratic form. The equation (1.
1)
of a
results when one considers the
matrices associated with the quadratic forms H(x, x) and P(x,
which are related in the following way:
dt
P(x, x)
where
dx
dt
=
xA
=
H (x, x)
,
x)
3
A
is a real
If we
nXn
matrix, and x is a real
1
Xn
row vector.
differentiate and make the appropriate substitutions, the
following equation results:
P(xA, x)
In
+
P(x, xA)
matrix form, equation
(xA)PxT
+
H(x, x).
=
(1. 2) can be
xP(xA)T
(1. 2)
expressed as
T
xHx,
=
or as
AP
P =[p..]
(i,j
=
+
PAT
1,2,...,n)
=
H.
and
(1.3)
H
= [
h..]
(i,
j =
1,2
...,n)
,
in
(1.3) are the symmetric matrices formed respectively from the
coefficients
transpose
of
of the
1.
and
P(x, x),
and
AT
is the
The central result of Lyapunov is
1.
The
positive definite
of
A
1
+
A
is stable (all of the
have negative real part) if and only if
matrix P
nXn
AP
Proofs for Theorem
real matrix
nXn
characteristic values
a
H(x, x)
the matrix A.
given in Theorem
Theorem
forms
PAT
=
-I
=
PT
exists such that
.
can be found in [ 10,
16]
.
4
We can
immediately generalize Lyapunov's work to the case
arbitrary complex matrix
of an
The quadratic forms
H(x, x)
A
a..]
= [
and
(i, j
=
1, 2,
,
n).
are replaced by the
P(x, x)
hermitian forms
n
H (x, x)
n
h,X.X.
i3 1
=
3
i=1
j=1
n
n
and
P(x, x)
p..13 x.1 x
=
i=1 j=1
Correspondingly, the matrix equation
(1. 3) is
replaced by the
matrix equation
H
where
H
= [
h. .]
i
(1,
=
j=
AP
+
PA',
1, 2,
,
n)
and
P
= [
P. .]
i
(i,
j = 1, 2,
,
n)
are the hermitian matrices formed respectively from the coefficients
of the
hermitian forms H(x,
x)
and
jugate transpose of A.
We follow
consider positive stable
A
P(x, x).
A''`
Ballantine
[
1, p. 2] in
rather than stable
A.
matrix is called positive stable provided each
of its
values has positive real part. We note that if
A
-A
is stable, and
-H
denotes the con-
An
that we will
nXn
characteristic
is positive stable,
is negative definite for positive definite
H.
5
If we
1
consider positive stable
rather than stable A,
A
can be generalized to complex matrices
Theorem 1*.
if and only if
The
nXn
AP
+
in the following way:
A
complex matrix A
a positive definite
PA*
=
matrix P
I
.
=
Theorem
P*
is positive stable
exists such that
6
II.
P- AP+ PA'''
SOME PROPERTIES OF THE MAPPING
Since we wish to characterize the set
AP
+
PA*
as
P
ranges over the positive definite hermitian matrices, it will at
times be more descriptive to utilize the mapping
[A]: P -AP+
PA *,
rather than the equation
H
=
AP+ PA*
P- AP
The fact that the mapping
by
specification
of the
of
A
notation [A]
= [
.
a,
ij
.
(i,
]
+
PA'
is completely determined
j = 1, 2,
for the mapping.
,
n)
justifies the use
(Since brackets are also
used in this thesis to enclose the components of a matrix and for
grouping, every effort will be made to make it clear from context
which meaning is intended.
the mapping
Lemma
of
order
list
a number of
properties
of
[A]
[
A
in the space
x
of
complex
is a linear transformation.
n
Proof: The mappings
Thus the mapping
We now
which we will call upon later in the thesis.
The mapping
1.
matrices
[A],
)
-AP and P -PA* are
P -AP + PA''` is linear.
P
]
:
clearly linear.
7
It follows
basis
x, [A]
of
matrix
from Lemma
of
A
[
product
]
determines an n2 X n2
[A] relative to this basis).
the ordered basis
by
that, with respect to each ordered
1
of
x
matrix (namely, the
We
will now show that if
is chosen properly, the matrix determined
where ® denotes the Kronecker
is
A ®I + I®A,
For
a definition and some of the
properties
of the
Kronecker product (which is also known as the direct or tensor
product) see [ 9, p.
81 -88]
or
[
2, p. 227
-229]
.
In each of the cited
references the authors found no need to define the Kronecker product
for nonsquare matrices. We will now define the Kronecker product
a row
vector by a column vector, since this concept will be used to
exhibit the particular basis of which we have spoken.
nX
1
of
column vector and
Kronecker product
z®y
y
a
1
row vector.
Xn
Let
z
be an
Then the
is defined to be the ordinary matrix product
zy. We shall now describe the above mentioned basis of X. Let e.
be an nX 1 column vector with 1 in the ith position and zeros
i
elsewhere. Similarly, let f.
the
jth
be a
1
Xn
row vector with
1
in
position and zeros elsewhere. Then
(2. 1)
where the matrix on the right side of (2.
1)
designates the
nXn
8
matrix with
in the
1
position and zeros elsewhere (this
(i, j)
matrix is sometimes denoted E..).
13
precede
{
e
.
1
e!
®f.: i
J
,
Q<)
j
for
V
J
=
1, 2,
,
If we
or for
require that e. ® f.
i <
i'
n
forms an ordered basis
}
i
=
and
i'
j'
j <
of
X
then
,
The
.
ordering described above is called standard lexicographic ordering.
To show that the matrix of
A
®I
+
IQ A,
[
A
with respect to this basis is
]
we will consider the mapping
individually. The mapping
respect to the basis
{
e. ®
P --AP
f.:
i, j
=
-AP
P
and P --PA
is linear and its matrix with
1, 2,
is determined by
n1
,
i
its action on the basis elements
(i,j
f.
1,2,,n).
=
have
j
We
column
^allal2
a21a22
a
a
_ nl n2
0-
aln^
i
all
-0
0._
ali
a2n
a
th
nn_
0
0
a
0
0
.
ms.
so that
ei®
f.
J
- k=1
Therefore the matrix
aki (ek® f.)
(i, j
-=
1, 2,
.
n)
.
J
of
P
- AP,
ordering the
{eX
1
f.J
:i,j
=
1,2,,n}
9
standard lexicographically, is
A
®
Proceeding in a similar
I.
P. PA*,
J.
fashion with the transformation
we find that a typical
basis element maps as follows:
n
ei®f.
® fk)
ak (ei
J
J
(i, j
=
2,
1 ,
. .
,
n)
.
k=1
Thus the matrix of this mapping with respect to the indicated basis
is
I®
le. ®f.:
J
[A]
Hence the matrix of
.
i, j
=
1, 2,
,
n
with respect to the
ordered standard lexicographically,
} ,
is
A®I+ I®Á.
Lemma
X. + X.
i
2.
(i,
If
A
j = 1, 2,
has characteristic values
(i= 1,2, '
X.
i
then
,n) are the characteristic values of A ®I+ I®A
J
and hence are the characteristic values of the mapping
Proof:
It is
well known that we can choose
that Al =PAP -1
diagonal
of
A1,
teristic values
is
, n),
of
is
triangular with
provided the
A
[
9,
clearly triangular with
characteristic values
calculations verify that
=
X. +
T.
J
(i,
1, 2,
1, 2,
,
Theorem 41.3, p.
X. + X.
i
(i
X.
=
A
]
.
nonsingular such
P
(i
X.
[
75]
.
n)
,
on the
n)
are the charac A1®I
+
I
©A1
on the diagonal and thus has
j = 1, 2,
,
n).
The following
J
Al ©
I +
I©
Al
is
similar to A©
I +
I®
A:
10
Al ®I
IQA1
+
=
PAP -1 ®I
I ©PAP -1
+
1
_
(P®P)(A®I)(P- ®P
_
(P®P)(A®I
+
-1)+
i®A)(P®P)-1,
(POP) is nonsingular. (The properties
since
1
®17)(I®A)(P- 017)-1)
of the
Kronecker
product used above are proved or references to their proofs are
given in
A®I
of
[
+
9, p. 82 -83]
are
I®A
Lemma 3.
If
A ®I + I®A
Proof:
Let
(i
I®A
that
+
=
,
Re (X.
+
X.)
(
X. + X.
for i,
(i, j
> 0
X.+
X.
i
of
Let
[A]
2,
,
n).
[A]
are nonsingular.
characteristic values
=
,n
j = 1, 2,
Re(X. +-X.)
j
(Lemma 2). We have
and therefore
,n)
1, 2,
A®I
+
I®A
means the real part of the com-
.
J
A
be any matrix of
all nXn hermitian matrices
Then
1 ,
positive stable matrix, then the
nXn
a
Lemma 4.
=
characteristic values
Then the characteristic values of
is nonsingular. Here
plex number
i, j
and thus the mapping
n).
are
I®A
+-X.
be positive stable with
A
1, 2,
.
is an
A
matrix
X.
X
It follows that the
. )
x
and let
47/4
be the set
is a subset of
(thus 90/
is a (real) linear mapping of 2Ì
into
x ).
and at
least one (real) matrix representing this real linear mapping is
similar (over the complex field) to
A
®I
+
I ®A
.
First, for
Proof:
for all real
Further,
[A]
to
P
V
maps
í7%
P'
=
r2.
and
r1
and
P1
Thus
implies
into PT
P2
/
in
r1P1
4
PA'``)'
+
r2P2
is in N
is a real linear space.
%;i
(AP
+
AP
=
so that
PA *,
+
The fact that the restriction of
.
matrices Ekk,
E
J
a
basis
)
(j
of
x
kj
,
k
,n,
1, 2,
=
form a basis
j < k)
are matrices
[A]:
x
-
x
+Ek.,
J
Thus the (real) matrix
(over the complex field).
of
k
'7/ and also
of
associated with this basis and the matrix
M[A]
a
and the real field being a subfield of the complex
x
One can routinely verify that the
i(E Jk -E
[A]
7,/ being
is a linear mapping follows from Lemma 1,
subset of
field.
11
.iI+ I® A
A
with respect to different bases and
hence are similar.
Before proceeding we make the following definition:
be a subset of a
r1R1
+
r2R2
real linear space. Then JR is
is in
positive (real)
.A for all
r2.
and
r1
and
R1
R2
definite matrices forms a cone. Further, if
nXn
(or any complex
let
H1
and
tive. Then
P2
in
kip
H2
H1
,
= [
matrix),
be in
A
]
P1
[A]
[ALP,
and
[
A
,,
A ] P2
r
and all
set iP of positive
is positive stable
is a cone.
;U"
and let
H2-
a cone provided
in
We note that the
Let ,R,
and
for some
To show this
r2
be posi-
Pl
so we have
r1Hi+r2H2
rl[A]P1+r2[A]P2
_
[A](r1P1+r2P2),
and
12
[A] JP,
which is in
since
,60
is a cone.
Since we are primarily interested in how
cone of positive definite matrices JP
[
A
] :
P
-AP +
definite and
we will call the mapping
,
P
be positive
be positive stable, the Lyapunov mapping.
Lyapunov mapping is thus
restricted to
maps the
subject to the restriction that
PA
A
[A]
[A]
(for some positive stable
and its range is
,
we are interested in
[
characterizing.
A
]
0.
[
We note
A ] X9
that both
the Lyapunov mapping depend on the positive stable
A
The
A)
is the cone
[A]
0)
and
under con-
sideration at the time.
We now
the cone
[
Theorem
2.
state three theorems which give information concerning
A] (P.
If
is positive stable and
A
H
is positive definite,
then
A P + PA
can be solved with
Proof:
H
=
RR's
We follow
P
(2.
H
2)
positive definite.
Taussky
[
13]
for some nonsingular
AP
which is equivalent to
=
+
Since
.
is positive definite,
Then we have (to solve)
R.
PA*
H
=
RR*
,
13
R- 1ARR -1P(R -1)yß+ R- 1P(R- 1)^RJA >4(R *)
=
I,
which can be written
A1P1
P1AI*
+
(2. 3)
I,
=
where
A1
=
PI
=
R
-1AR
1P(R
,;.
Now since
Al
values as
A),
with
positive definite. Hence it follows that
P1
is positive stable (it has the same
it follows from Theorem l''` that (2.
positive definite and satisfies (2.
Theorem 3.
characteristic
H
=
positive definite
AP
+
P
PA
P
3)
=
can be solved
RP1R*
is
2).
for some positive stable
if and only if
A
and some
has at least one positive
H
characteristic value.
Stein gives a proof of Theorem
Functions
of
and Theorem
3
in his paper On the Ranges of Two
Positive Definite Matrices
3
[
11 ]
.
From Theorem
2
it follows that the range of the Lyapunov mapping
contains the cone
of
positive definite matrices but does not contain
any nonpositive definite matrices.
In Lemma 3, when we
established
14
that
[A]
was nonsingular for positive stable
the existence of
[A] -1.
characterize the range
This existence of
[
A
]
)P
A,
[A] -1
we
assured
allows us to
of the Lyapunov mapping in the
following way.
Theorem
H
4.
is in
In the next
Let A
[A]
0
be positive stable and
if and only if
[
A
]
-1H
H
be hermitian. Then
is positive definite.
section (and in later sections) we use Theorem
characterize the range
of the Lyapunov mapping for some
positive stable matrices.
4
to
special
15
III.
In this
[
A ],a) FOR DIAGONAL
A
section we consider only positive stable matrices
which are diagonal.
A
These matrices display their characteristic
values, and since [A]
and [A] -1H
P
are easily calculated,
characterize the image cone [A]6).
we can use Theorem 4 to
The
simplest kind of diagonal matrix is a scalar matrix, which is just a
scalar multiple
case where
tion that
of
A
is a positive stable scalar matrix
A
H
0
[A]
rather
a
be hermitian.
i. e.
,
Then
A
Re(X)
Ps
A] P
P(XI)*
= [
if and only if
(XI)P
+
=
XI
restric-
> 0,
be positive stable
and so
A
is in
H
,
_ (X
is positive definite (since
Thus for scalar
is a cone).
itself by [A]
H
=
The
XI.
severe one, and the results are indicative
if and only if, for some
H
the cone
S
X
+ X > 0
and
is mapped onto
.
is an
When A
of
is
XI
=
identity. For this reason we first examine the
the severity of the restriction. Let
and let
a)
of the
nXn
nonscalar diagonal matrix, the range
the Lyapunov mapping is not so conveniently described.
considering the
case in detail.
it is of the form
nXn
A
2 X 2
Before
case, it is interesting to examine the
2X2
complex matrix is hermitian if and only if
16
w
x+iy
L-iy
where
the
are real and
w, x, y, z
2X2
=.
i
If we
fix the basis (for
hermitian matrices) to be the one mentioned in Lemma
4, with the ordering
2X2
J
E
,
E
12
+E
21
i(E
,
hermitian matrix is associated with
12
-E
21
), E
then each
22,
a 4 -tuple by the following
mapping:
w
x+iy
x-iy
z
(w,x,y,z),
which is an isomorphism from the space of
to the space of real 4- tuples.
2X2
Now if we apply the
space of 4- tuples, we can describe the cones
geometrically since they are subsets
matrices. First, a point
(w, x, y, z)
hermitian matrix) corresponds to
matrix
usual distance to the
and
set of
of the
[
2X2
(this represents a
A] W
hermitian
2 X 2
positive definite (hermitian)
if and only if
w > 0
(We get
a
QP
hermitian matrices
and
wz - (x2 +y2) >
from the above conditions that
0
.
z > 0
also.
)
The set of
points satisfying this system of inequalities forms the interior of
17
(one nappe of) a four -dimensional cone.
The traces of the boundary of
this cone in each of the three -dimensional coordinate hyperplanes
(a c oor -
dinate hyperplane is the set obtained by setting one of the coordinates
equal to zero and is labeled by the remaining coordinates) are given
by the following:
(1)
If
x
=
we have the
0,
w,
hyperplane.
y, z
The condi-
tions which describe the boundary of the cone in this hyperplane are
and
w > 0
wz -y2
=
0.
This is an elliptic cone of one nappe with axis
is tangent to the
positive
w
(2)
If
plane and the
wy
and
z
y
0,
=
y
=
w
-z
= 0
and which
plane respectively along the
yz
axes.
we get the
hyperplane. The condi-
w, x, z
tions that define the boundary in this hyperplane are
and
w > 0
wz
2
=
0,
and the locus is similar to the one described in (1).
(3)
or
line;
If
w
= 0
or
=
0,
we get the hyperplanes
x, y,
z
The trace in either of these hyperplanes is just a half -
w, x, y.
the positive half
half of the
z
w
of the
z
axis in the latter.
axis in the former and the positive
18
[A]
Now if we apply the mapping
to the cone
lU"
of
posi-
tive definite matrices, Lemma 4 assures that the matrices in the
[A] a) will be hermitian. Thus the image can be described
image
geometrically in the same manner as a)
A
2X
2
diagonal matrix is positive stable and nonscalar if
and only if it is of the form
and
Re(µ)
> O.
Thus let
w
P
X
0
0
µ
and
with
X
Re(X) >
0
this form and let
a
b+ic
b-ic
d
I¡'
/ µ,
H=
z
hermitian. Then
_
A be of
x+iy
=
x-iy
be
.
is in the range of the Lyapunov mapping if
H
and only if
P
is positive definite.
Lemma 3.
a
H
With
=
[A] -1H
The existence of
A, P and H
b+ic
X
0
w
d
0
µ
x-iy
[A]
-1
is guaranteed by
as given we have
x+iy
w
x+iy
z
Lx-iY
z
X
0 1
0
µ
=
b-ic
(\+
.
)w
(µ +X)(x -iy)
From this we get
(X
+µ) (x +iy)
(µ +µ )z
19
b+ic
a
+µ
X+-X'
x±iy
w
P
=
=
x-iy
[A]
H,
z
b-ic
d
µ+
and it follows that
µ+µ
is in the range of the Lyapunov mapping if
H
and only if
X
a>
+ X.
0
ad
and
b2
(R+T)(µ+µ)
Thus the point with coordinates
hermitian matrix in [A]
a > 0
and
XP
(X
c2
+
> 0
+µ)(µ +X)
corresponds to a
(a, b, c, d)
2X2
if and only if
Kad -(b2 +c 2)
> 0
,
where
K
(X+1-0(X+11)
(X+\)(+)
The following calculations verify that
every
?
K >
)
1
is of this form (with
K >
Re(X)
1
and conversely that
> 0,
Re(p.)
> 0
and
Namely, we have that the following inequalities are pair-
wise equivalent:
20
2>
IX-p.1 >
0
µ+µ
XX + 1111>
1X +
µµ
+-X
+ X11 > X11 +
(X
+µ)(X+µ)
> (X +
(X
+µ)(µ+X)
>
(X
+fi)(µ +µ)
The conditions which
H
µ).
+
11
Xµ
+
)(µ+µ)
1
must satisfy to be in the range of the
Lyapunov mapping resemble those conditions a matrix
satisfy to be in 6)
K
For
scalar) the cone
= 1
(i.
e.,
have already seen.
must
the one difference being the coefficient K.
,
Therefore, how this
K
P
A
affects the cone is
of
prime interest.
[Al
To see what happens as
as we
_63%
gets large, we
K
shall look at the traces of the boundary locus in the respective
coordinate hyperplanes.
When
a
or
d
equals zero,
not affect the locus and thus it is the same as that of 62
corresponding hyperplanes. If we let
boundary in the
a, b,
d
c
of
hyperplane, the effect of
this trace are the following:
a >
0
and
Kad -b 2 -
O.
does
in the
and consider the
= 0
The necessary and sufficient conditions for
K
H
K
can be seen.
to be on the boundary
21
It is
clear from the equation Kad -b2
positive definite.
any
As
H
such that the diagonal entries of
are positive we could find an
such that
of the
in
[
A
(2X2
similar locus arises when
A
contains the cone
A,
when
When
-
(2)
n
= 2
is a cone which
and whose size depends
is diagonal, in the manner just described.
A
n >
for all diagonal A
VP
(1)
1:
is set equal to zero.
b
Thus the range of the Lyapunov mapping when
on
be in the range
H
corresponding Lyapunov mapping (see Lemma
1, p. 3] ).
H
positive stable and diagonal)
was large enough to assure that
K
need not be
H
increases, b2 /(ad) must increase. For
K
hermitian
(2 X2)
that
= 0
it is no longer necessary that the
2
characteristic
values be distinct for the matrix to be nonscalar. In general a diagonal positive stable matrix A
where the
Re(X.)
(i
> 0
=
can be denoted by
1, 2,
,
are not necessarily distinct. Now if
is hermitian and
P
= [
p..]
(i, j
=
H
1, 2,
AP
=
"
` ,
+
n),
PA
n)
H
and the
=
[h..]
ij
diag
X
(X1,X2,
.(i =1, 2,
(i, j
= 1,
2,
for some positive definite
we have
,
,
,
n)
°, n)
Xn),
22
hllh12.
hln
h21h22
h2n
hnlhn2
hnn
..
0
A2
0
p
X
0
0
0
X
n
+-X
(X.
h..
3.3
Hence the
(i, j)
_
P21P22
p2n
Pn1Pn2
pnnr
p1 n
P21P22
P2n
Pn1Pn2
Pnn
-
-
2+-X.
pin
pl 1p12
(X1 +X.1)p11
(X
pllp12
1)p21
)pn1
(X.
(X
1
+X2)p12
...
+X. 2)p22
..
(ñn +X2)pn2
...
X
0
0
X2
0
0
(X.1
(X2
-
.
0
...
0
X.
n
_
+Xn)pin
+Xn)p2n
(Xn +X
)pnn
n)
(..+X..)p13
(i, j
entry
positive definite matrix P is given by
i
.
=
1, 2,
J
of the
23
pij
1J
-
j)
(X. +
=
-1
h..
1J
J
i
(i, j
=
1, 2,
.
n).
.
One of the well known conditions for positive definiteness applied to
would give a system of requirements that
P
H
must satisfy
to be in the range of the Lyapunov mapping.
To get more information we now change the notation as follows:
let
X.
(so
k < n).
i
(i
1, 2,
=
,
Then
characteristic values
be the distinct
k)
is
A
similar to
a
A
of
block diagonal matrix, each
diagonal block of which is a scalar matrix, with distinct blocks cor-
responding to distinct scalars. Thus we may assume
in this form.
(i
P
=
1, 2,
and
,
H
Let the
k,
where
diagonal block of
m.
i
->
for
m.
J
H
P.
,
1J
is in the cone
[A
H..
=
for some positive definite
are each
]
kP
(i,
(i,
j = 1, 2,
dimensions
,k)
m. Xm..
1
if and only if
j = 1, 2,
Note that the
P.
of
m.
and let
)
.
j)Pij
J
iJ
(X. +
i < j
l
H..
and
1J
have order
A
P.131.
and H.J
il
be partitioned into blocks
respectively, where
We find that
ith
itself is
A
m1
.
X
,
k)
m1
leading
principal submatrix
H11 -
is positive definite, since
)P11
(X1 +
1
X1
+
X1
> 0
and
P11
is the
leading principal submatrix of a positive definite matrix.
mXm
It is
easily
J
24
verified that thus
H
has at least
values. We recall that ml
m1
>
-
m.
theorem
(i
of
1, 2,
=
,
positive characteristic
m1
is the multiplicity of
N.1
and that
This result is a special case of a
k).
P. Stein and A. M Pfeffer.
That theorem is given
next without proof in the present author's wording.
Theorem
Let A
5.
be an
positive stable matrix with
nXn
m
equal to the maximum number of Jordan blocks associated with
one
characteristic value.
If
H
has at least
m
with at least
similar to
A
m
P
=
is positive definite, then
AP
+
PA*
positive roots; further, for every hermitian
H
positive characteristic values there exists a
B
and a positive definite
H
P
=BP+ PB*
such that
.
2/
Mentioned by O. Taussky in an address at the Argonne Matrix
Conference March 30 - 3 1, 1967; to appear soon in the Bulletin
of the International Computation Centre.
25
[A I
IV.
FOR A GENERAL
In this section we use
[A161,
when
Theorem
this case we hoped that
A
to characterize the cone
general positive stable matrix.
is a 2X 2
A
4
2X2
[A]-1H might be familiar and thus
might be able to generalize our results to larger
were unable to see any such generalization
and thus we feel that Theorem
characterizing the cone [A ]
4
of the
In
we
(n > 3).
n
We
results obtained,
is limited as a practical method of
in the general nXn
W
case if
n > 3.
We now
ated with
matrix.
[
A
proceed to find the inverse
]
where
,
is a general
A
4X4
of a
2X
2
matrix associ-
positive stable
Let
a
A
=
d
c
be a positive stable matrix.
the basis
{
e.
®
f,
:
i,
The matrix of
j = 1, 2,
,
n
}
[ A. ]
(ordered standard lexicograph-
J
ically) then will be
A®
I
+
I® A
with respect to
a+a
b
b
0
c
a+d
0
b
c
0
d+a
b
0
c
c
d+d
=
26
We will use the adjoint method to find the
inverse of A® I
+
I®A.
The adjoint matrix is defined to be the transpose of the cofactor
matrix, and thus
[AQI+ I© A]
The factor
ad -bc
-1
=
= [
det(AQI
+
IQA)]
adj(A® I
occurs in many
det A
of
I®A)
+
the components of
the adjoint matrix; therefore, we will use the substitution
A
also find that expressions
We
ad -bc.
=
of the
form
A -L
occur frequently.
Thus a substitution of the form A
transpose
such that
A'®
I© A'
I +
siderably.
a
and
d'
=
A)
of
d
=
A ®I + I ®A,
We note
E
A'
=
det A'
AQI + IQA,
FaT
b
c
d'
so we let
a'
=
a -E
and
Then set
=
a'd'-bc. Before solving the equation
=a+d+a+d
(not the
is real and such that
A'
=
we let
T
A'
would also simplify the adjoint con-
is pure imaginary.
A'
so that
=
-
that subtracting the same imaginary number from
does not change
where
d -E,
0'
A(a +d) +L(d +a)
and
=trA+trA
= X
+
µ
+X+µ>
>
A'
for
E,
27
where
since
D'
=
and
X
of
A.
Now,
is pure imaginary we find by routine calculation that
E
A'
are the characteristic values
µ
only if
E
-
A-0
T
(which is obviously pure imaginary).
A
Conversely, if
T
then the same calculation shows that
E,
A'
=
A'
Using this value for
.
we see that indeed
A'
and hence that
®I
[
A'
+I
®A'
=
[A]
]
=
A
a..
(i, j
=
+I®A
We will now
.
components of the adjoint of A®
denote them
®I
1, 2, 3,
repeatedly use the fact that
A'
I + I®A
(=
A' © I
+
I ®A'
)
and
(During these calculations we
4).
=
determine the sixteen
A'
To simplify the notation
)
.
during the calculation we will drop primes until otherwise specified.
all
-
_
a+d)(d+a)(d+d)
-
bc(a+d)
-
(ad-bc)(a+d)+ (ad-cb)(d+a)
(a+d)
_ (A
+ A
+dd)T
(d+a)
+
dd(a+d
+
cb(d+a)
+
dd(a+d
a+d)
+
a+d)
28
a12
{b[ (d+a)(d+d) - cb]
= -
=
-b[ (d 2+ad + ad + dd - cb
=
-bdT
cb)
+
ad
-
ad]
=
b(-cb)
,
=b[ -cb
a14
+
}
-0+d(d+a+a+d)]
_
a13
cbb
+
-
-
(a+d) b(d+d)
(ad
+
+ad+dd+d2+cb)+ad
b[
=
-bd
=
-[ -bb(d+a) - (a+d) bb]
=
bbT.
-
+
-
ad]
A+
=
T
cb2
,
The remaining components are found in analogous fashion.
simplify to the following
a21
=
a22
=
a23
=
a24
=
-cdT,
(/A+ ad)T
bcT
-abr
,
,
:
They
29
=
-cdT,
a32
=
cbT,
a33
=
(ad+A)T,
a34
=
-abT,
a41
= CCT,
a
=
-caT,
=
-ac T,
a
a
31
42
43
a44
If we
=
(A+
denote the determinant of
D
We can see
Re(µ)
aa)T
> O.
= (X +
from
X)(X +
(4. 1)
that
A ®I + I ®A
µ)(µ
+
X)(µ+
by
we have
D,
(4.1)
).
is positive since
Re(X)
> 0
Thus the inverse of the matrix A ©I+ I©A
is
D
A+ dd
-bd
-db
bb
-cd
A+ad
bc
-ab
-cd
cb
A+ad
-ab
cc
-ca
-ac
T
D
and
A+aa
amp
and we can now find
[A]
1H.
To do this we
order
H
(standard)
30
lexicographically, getting a
perform the operation (AOI+ I ©A)
We can now
H.
column vector which we denote
4 X1
rearrange the result in the usual
given by the following,
when H
matrix form.
2X 2
and then
1H
[
A
]
-1H
is
w
=
z
ú
w(A+dd) - ubd --u-db
+
zbb
u(A+ad)
-
wcd
-
zab
+
ucb
(A+ad) - wcd - zab
+
ucb
z(A+aa)
-
uca
-
uca
+
wcc
T
D
u-
It
turns out that
[A
where
A
[
A
H
]
]
-1
can be written in the form
H
D (OH +
=
*
AHA'),
(4. 2)
is the transpose of the cofactor matrix of
A.
We note
that
(Á
,.
-dc
-b
rd
-c
rd
a
-b
a
-(A*),
and thus there should be no confusion in the expression (4. 2).
this point we restore the primes and consider the conditions
must satisfy to be in the cone
[
A
]
1E1
[A
]P
.
(4. 2)
D (A'H
=
D
=
ti
D{(A'-E )H+A.HA"+EAH
{
A'H
H
becomes
A.'HA' *)
=
+
At
+
(A-EI)H(A'+EI)}
- EHA
.
(4.3)
31
Thus
H is in
definite.
the cone
[
A
]
pif and
only if (4.3) is positive
32
CONDITIONS ON H AND A
TO HE[A]a)
V.
Using the equation
H
=
AP
+
EQUIVALENT
or the related mapping is
PA*
not the only method one might consider to characterize the cone
[
A]0).
In this
one might use to
section (Theorem
four equivalent methods
6) we give
characterize the cone. Some motivation for these
is as follows:
The integral formula in (2) (in Theorem
Bellman
[
2,
Theorem 6, p.
175]
,
6)
is essentially given by
where he assumes the existence
integral, but later fails to discuss that existence as he promises.
of the
The equation in (3) is one used by Stein
[ 11 ]
and by Taussky [13]
in a manner very close to that used here.
motivated by an exercise in Bellman
(4) was
Lemma
5
2, p. 99, Ex. 7]
Qp
by means of its dual cone
all nonzero nonnegative definite matrices
of
of (1) and (4)
means
of
In (5) we
Our
.
is a corrected and simplified version of that exercise, and
characterizes the cone
set
[
(in Theorem 6)
its dual cone
([
A* ] - 1É9'
assume
[A
A
),
the
The equivalence
similarly characterizes
simply observe that
In this section we
B.
(6)'
[
A]
a) by
)
1]
i
and
=
H
[Au.
are given.
We now
33
state and prove Theorem
Theorem
6.
Let
H
5.
be an
nXn hermitian matrix and
positive stable matrix.
nXn
A
be an
Then the following five statements are
equivalent:
(1)
H
=
(2)
AP+ PA*
e
for some positive definite
AtHe- A`tdt
is positive definite
P;
;
S0
(3)
C
(4)
H
=
=
(A+I) -1(A - I)
tr(KH)
A *K +KA
(5)
P- CPC* for some positive definite
H
Proof:
=
where
;
for every nonzero hermitian
> 0
P,
is nonnegative definite
A 1P + P(A
We will show
1)*
that
such that
;
for some positive definite
P.
equivalent to each of the other four
(1) is
statements. To show that
(1) and (2)
inte g
in (2) is
the integral
-114,
[A]
K
are equivalent
we will show
that
and thus equivalence follows from
Theorem 4. To verify this let
P
=
s ° e -AtHe -A
*
t
dt,
(5. 1)
0
where the exponential function
of a
matrix is given by the usual power
34
series, which always converges
Theorem
2,
matrix such that Al
be a nonsingular
Jordan blocks
[
of the
=
2, p. 166]
1AS
S
.
Let
S
is the direct sum of
form
n
0
..
.
0
1
0
.
1
1
=
Then to show that (5.
1)
.
X.
1
Lo
AI+R.i
i
exists it is sufficient to show that the integral
co
P1
=
exists, where
(c. f.
[
H1
=
S
{-Alt}
exp {-Alt}H1 exp
`
`J
dt
0
1HS
1
I\,
since
4, Theorem 10. 2, p. 196])
exp
{
-(SAS- 1) }= S(exp{ -AIDS
and hence one easily sees that the
given integral equals
co
S
1
exp {-A.lt}H1
exp(-
and that
SH 1S*.
y
m
t}dt S',
JO
i. e.
,
P
=
SP 1S
H
=
Let
H1
be
partitioned
in such a way as to make the block multiplication
exp
{
-Alt
}H1 exp
{
-Al t
}
defined
Then the
(i, j)
block of
Pl
1
35
will be given by
oo
(P1)i3 -
g0
('
=
exp{-(XiI+Ri)t } (H1)iexp{-(.J I+RJ^ )t } dt
co
-
exp { - (Ai+j )t } exp { -R it
`
}
(H 1L.exp
{
-R t } dt
.
0
Note that
R.
and
are nilpotent, so the exponentials involving
R.
them are polynomials in t.
blocks of
vergence
and thus the existence of
P1
of
Consequently, the existence of the
integrals
°0k
t exp
J0
{
follow from the con-
form
of the
,
P
-(?.+.)t
}dt.
1
Churchill in [ 3, Theorem
J
1,
absolutely convergent for
p. 171] proves that this
k > 0
and for
Re(X. +X.
integral is
)
> 0
Since
.
3
we know that the
stability
of
A
guarantees the uniqueness
of
P
(Lemma 3), all that remains is to show that the integral formula does
in fact yield the solution.
Calculating, we have
36
oo
AP
+
PA
=
5(Ae -At He-A
e-AtH -A >tA
t
e
)dt
0
d
_
s:-
=
lim
T
=
We used the
e
(
dt
-
[
AtH
e -A t ) dt
e-AtH e-Ay4t]
0
co
H
T
.
fact that
e
-AT
which is verified in
-i
[
as
0
2, p. 241
We next show (1) is
T
--
oo,
-2421.
equivalent to
(3)
using the equivalent
transformations
(A - I)
C
=
(A + I) -
A
=
(I-C) 1(I+C)
1
and
.
To show that (1) implies (3) we need only substitute for
terms
of
C
by the above
transformation, and we get
A
given in
37
H
=
(I-C)-1 (I+C)P
+
P(I+C `)(I-C
-1
r)
1
where
(I-C)P (I+C*)] (I-C*)
=
(I-C)-1[ (I+C)P(I-C*)
=
(I-C)-1[ 2P
=
2(I-C)-1P(I-C F)-1-C 2(I-C)-1P(I-Cry4)-IC*
+
2CPC*] (I-C>;.)-
-
=P1
-CP1C,
P1
2(I- C) -1P[ (I -C) -1]
=
fact that
We used the
C
1
and thus is positive definite.
commutes with
assume
for some positive definite
(3)
implies
P
and make the substitution given above.
H
(1) we
=
P
=
(A +I)
=
(A +I)
-
(A+I)-1(A-I) P
P1
=
=
P -CPC
This yields
(A `-I)(A ' + I )
-1[ (A +I)P(A +I) -(A -I) P (A -I)] (A *+I)
1
[
2AP+ 2PA.*]
=AP1 +PlA
where
H
(I-C)-1. To show that
1
(A.,, +I) -1
,
2(A +I) 1P[ (A +I) -
and is thus positive definite.
]
Before we establish the equivalence of parts
(1) and (4), we
state and prove an essential lemma.
Lemma
5.
Let
P
be an
nXn
hermitian matrix. Then P
positive definite if and only if tr(PB)>
0
for every nonzero
is
38
nonnegative definite (hermitian)
Proof:
tr[
,,
(T
We
B.
first note that for nonsingular T,
PT)B]
=
tr[ P(TBT''`)]
only consider diagonal
Thus in proving the lemma we need
.
P,
since every hermitian matrix is con-
junctive with a diagonal matrix, and a matrix is nonzero nonnegative
definite if and only if every matrix conjunctive with it is nonzero
nonnegative definite. Let
definite, and
B
= [
b..]
iJ
definite. Then
X..
> 0
P
(i, j,
(i
=
i
with
b..u
>
=
diag
1, 2,
=
1, 2,
for at least one
O
(X1, X2,
n),
be positive
Xn)
be nonzero nonnegative
n)
,
,
,
b..ii
and
-> 0
(i= 1,2,
,n)
Thus
i.
n
tr(PB)
=
)
L
X.b.. >
1 ii
0
1=1
for any nonzero nonnegative definite
B.
for every nonzero nonnegative definite
(i, j, k
= 1,
2,
'
,
n)
be the n
1
0
Then the
Bk
Now
B.
assume that tr(PB)>
Let
Bk
matrices with elements
if
i
= j =
k
otherwise
are nonzero nonnegative definite and
tr(PBk) = Xk
(k
= 1,
2,
,
n)
.
bi'J , k
0
39
Therefore
for
Ak > 0
k
,n,
1, 2,
=
and it follows that
P
is positive definite.
We
are now ready to show that
we must show that
such that
tr(KH)
for some positive definite
Note that
P.
for some positive definite
tr [ K(AP
tr
=
To do this
for every nonzero hermitian
> 0
and thus is nonzero for all nonzero
=
implies (4).
is nonnegative definite, whenever
A K + KA
tr(KH)
(1)
(KAP
+
P.
K.
A
H
=
is
KA
Suppose that
K
AP +PA*
K
H
=AP +PA
Then
PAN
+
KPA
*
)
=
tr(KAP)
+
tr(KPA
=
tr (KAP)
+
tr
(A
)
KP)
=tr[(AK+KA)P]
(Lemma
> 0
whenever
(4)
A K + KA
implies (1),
such that
Then
H
let
A K
+
A
P,
= [
]
is nonzero nonnegative definite.
tr(KH)
KA
5)
> 0
for every nonzero hermitian
is nonnegative definite.
and so
To show that
Let
P
=
[A]
K
1H.
40
tr(KH)
=
tr[ (AP
=
tr (APK)
+
tr (PA
=
tr(PKA)
+
tr(PA K)
=
tr [ P(KA
+ A JFK)]
=
tr[ P(A *K
PA*)K]
+
+
KA)]
K)
.
For every nonzero nonnegative definite
hermitian)
AK
+
KA
namely
K,
=
[A ]K
tr(PB)
=
=
K
[A*] -1B,
a (nonzero
such that
and hence such that
B,
tr[ P(A*K
It follows from Lemma
=
there exists
B,
5
+
KA)]
that
=
tr(KH)
P
> 0
.
is positive definite, which com-
pletes this part of the proof.
To show (1) implies (5), let (1) be given.
H
=
AP
=
(A - 1A)
=
where
if we
+
PA
PA
A-1P1
P1
=
assume
Then
APA
+AP(AA'-1)
P1(A-1),*.
and is therefore positive definite.
(5) is given, we have
Similarly,
41
H
=
A- 1P
=
A-1P[ (A,.)-lA*]
P(A
+
=AP1+ PIA
where
(5)
P1
implies
=
A
-1
P(A
)
*
+
(AA-1)P(A-1)*
',
-1
)
*
and is thus positive definite. To show that
(1) we could have
applied the part already proven ((1)
implies (5)) to the positive stable matrix
proof of Theorem
5.
A
-1
.
This completes the
42
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