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Remarks on two recent results in matrix theory
LINEAR
ALGEBRA
AND
ITS
APPLICATIONS
6, 109-112
Remarks on Two Recent Results in Matrix IRVING
JACK
The George
Theory
KATZ
Washington
Washington,
109
(1972)
University
D.C.
Communicated
by H. Schwerdtfeger
1. INTRODUCTION A complex conjugate original
n x n matrix
transpose formulation
known
the same
of the definition
111 that A is EP,
an Y x r nonsingular
A of rank Y is called EP,
of A, have
Let At denote the generalized
THEOREM
if
and only
THEOREM
D o
U”
0 o
purpose
matrix
U and
1
U.
matrix
recently
A.
obtained
the following
matrices.
results.
The eroduct AB is EP,
A and B have the same column spaces.
2. If A, B, and AB are EP, matrices,
results of Boros,
It is
inverse of A (see [5]) and R(A) the column
1. Let A and B be EP,
if
i.e., the “reverse The
[l]
[6] for the
D such that
of A for any rectangular and Katz
(see
as given by Schwerdtfeger).
[
Baskett
if A and A*, the
if and only if there are a unitary
matrix
A=
space
null spaces
order law” of this note
then (AB)t
= BtAt,
holds. is to show that
[2] and of Schwerdtfeger
the following
two recent
[7] follow easily from Theorems
1 and 2. Copyright
0
1972 by American
Elsevier
Publishing
Company,
Inc.
IRVING JACK KATZ
110 Let
THEOREM 3 (Boros,) . A = P* be EP, matrices; matrices.
D
0
0
0
1 P,
P, Q aye unitary matrices, and D, E are
The product AH is EP,
if
ifthere
and only
Y x
Y
nonsingztlar
is a unitary matrix T
such that TP” where K, M aye Y x Y nonsingular
=
matrices, and F o
PQ*=
1
0 G ,
[ where F is an r x Y nonsimgular matrix.
If A, B, and AB
are EP,,
the
reverse order law holds.
THEOREM 4 (Schwerdtfeger).
Let A and B be any complex rectangular
gnatrices of the same rank $ for which the Product matrix BA can be formed. If the eigenvectors for the nonzero eigenvalues of the two matrices AA* B*B
@an
and
the same space, then the reverse order law holds.
2. PROOF OF THEOREM 3 Suppose that A, B as given and A B are EP,. By Theorem 1, A and the same column space. In 11, Section 31 (in particular, p. 93),
B have
it was shown
that,
B is an EP,
matrix
where
if
and A and B have
C is an Y x r nonsingular
matrix. F u
PQ*=
[
the same column
Let
1
S G >
space,
then
ON
TWO
RECENT
RESULTS
where F is an Y x
thus
MATRIX
by P and postmultiplying
and hence CS = 0, UE = 0.
If we set T = P, then
form and
TP*
mentioned 3.
is immediate
Let
W denote
Theorem matrix
12.21)
from the representation
subspace
of AA*.
that
block
diagonal
diagonal
form.
of EP,
matrices
4
the
eigenvalues
block
1.
PROOF OF THEOREM
nonzero
TQ* has the required
= I also has the required
in Section
by Q* yields
Thus S = 0 and U = 0, and so F is non-
singular.
The converse
111
THEORY
Now
matrix.
7
premultiplying
IN
spanned
by
the
It is well known
W = R(AA*).
eigenvectors
for the
(see, for example,
It is easy to check
that,
[4,
for any
C
ct* =
C”f,
= R(U),
R(C”)
R(CX)
= qq.
Thus R(AA”)
= R(A)
= R(A*t)
Then AA*,
AA?, are EP,
[l,
31):
Section
AA”
over,
AAt
R(B*B),
matrices
D
= U* i 0
where U is a unitary
matrix,
is idempotent
= R(A*tA*)
0 0
= R(At*A*)
with the same column space so (using
1
U,
AAt
E U” [ o
=
D, E are p x p nonsingular
so E = I.
From
B*B = U” matrix.
0 o 1 U, matrices.
the hypothesis,
so
where F is a p x p nonsingular
= R(AAT).
U, Then
More-
R(AA*)
=
112
IRVING
AA'f'B*BA=U" Similarly
JACK
KATZ
[t :],:,:*[:,~]ZJA=B*BA
one proves
BfBAA"B*=AA*B*. From a theorem
of Greville
[3, Theorem
11, the reverse order law follows.
REFERENCES 1 T. S. Bask&t
and I. J. Katz,
2 E. Bores,, An.
Univ.
x2777. 3 T. N. E. Greville, 4 H. L. Hamburger Vector
Space,
Timisoara SIAM
Penrose,
6 H.
Schwerdtfeger,
Proc.
7 H. Schwerdtfeger, June
Rev.
Introduction
1969;
Sot.
to Linear
Linenr Algebra version
(2) 2(1969),
8(1965) [Math. Reviews
Transfovmations
Press, London
Philos.
Groningen,
revised
and Appl.
87-103. 37(1969j,
518-521.
8(1966),
University
Cambridge
2nd ed., P. Noordhoff,
Received
Algebra
and M. E. Grimshaw, Linear
Cambridge
5 R.
Linear
Ser. St;., Mat.-Fix.
51(1955), Algebra
received
and
1(1968), October
1956.
406-413. the
Theory
1962. and A$@.
in n-Dimensional
and New York,
325-328. 1969
of Matrices,
ALGEBRA
AND
ITS
APPLICATIONS
6, 109-112
Remarks on Two Recent Results in Matrix IRVING
JACK
The George
Theory
KATZ
Washington
Washington,
109
(1972)
University
D.C.
Communicated
by H. Schwerdtfeger
1. INTRODUCTION A complex conjugate original
n x n matrix
transpose formulation
known
the same
of the definition
111 that A is EP,
an Y x r nonsingular
A of rank Y is called EP,
of A, have
Let At denote the generalized
THEOREM
if
and only
THEOREM
D o
U”
0 o
purpose
matrix
U and
1
U.
matrix
recently
A.
obtained
the following
matrices.
results.
The eroduct AB is EP,
A and B have the same column spaces.
2. If A, B, and AB are EP, matrices,
results of Boros,
It is
inverse of A (see [5]) and R(A) the column
1. Let A and B be EP,
if
i.e., the “reverse The
[l]
[6] for the
D such that
of A for any rectangular and Katz
(see
as given by Schwerdtfeger).
[
Baskett
if A and A*, the
if and only if there are a unitary
matrix
A=
space
null spaces
order law” of this note
then (AB)t
= BtAt,
holds. is to show that
[2] and of Schwerdtfeger
the following
two recent
[7] follow easily from Theorems
1 and 2. Copyright
0
1972 by American
Elsevier
Publishing
Company,
Inc.
IRVING JACK KATZ
110 Let
THEOREM 3 (Boros,) . A = P* be EP, matrices; matrices.
D
0
0
0
1 P,
P, Q aye unitary matrices, and D, E are
The product AH is EP,
if
ifthere
and only
Y x
Y
nonsingztlar
is a unitary matrix T
such that TP” where K, M aye Y x Y nonsingular
=
matrices, and F o
PQ*=
1
0 G ,
[ where F is an r x Y nonsimgular matrix.
If A, B, and AB
are EP,,
the
reverse order law holds.
THEOREM 4 (Schwerdtfeger).
Let A and B be any complex rectangular
gnatrices of the same rank $ for which the Product matrix BA can be formed. If the eigenvectors for the nonzero eigenvalues of the two matrices AA* B*B
@an
and
the same space, then the reverse order law holds.
2. PROOF OF THEOREM 3 Suppose that A, B as given and A B are EP,. By Theorem 1, A and the same column space. In 11, Section 31 (in particular, p. 93),
B have
it was shown
that,
B is an EP,
matrix
where
if
and A and B have
C is an Y x r nonsingular
matrix. F u
PQ*=
[
the same column
Let
1
S G >
space,
then
ON
TWO
RECENT
RESULTS
where F is an Y x
thus
MATRIX
by P and postmultiplying
and hence CS = 0, UE = 0.
If we set T = P, then
form and
TP*
mentioned 3.
is immediate
Let
W denote
Theorem matrix
12.21)
from the representation
subspace
of AA*.
that
block
diagonal
diagonal
form.
of EP,
matrices
4
the
eigenvalues
block
1.
PROOF OF THEOREM
nonzero
TQ* has the required
= I also has the required
in Section
by Q* yields
Thus S = 0 and U = 0, and so F is non-
singular.
The converse
111
THEORY
Now
matrix.
7
premultiplying
IN
spanned
by
the
It is well known
W = R(AA*).
eigenvectors
for the
(see, for example,
It is easy to check
that,
[4,
for any
C
ct* =
C”f,
= R(U),
R(C”)
R(CX)
= qq.
Thus R(AA”)
= R(A)
= R(A*t)
Then AA*,
AA?, are EP,
[l,
31):
Section
AA”
over,
AAt
R(B*B),
matrices
D
= U* i 0
where U is a unitary
matrix,
is idempotent
= R(A*tA*)
0 0
= R(At*A*)
with the same column space so (using
1
U,
AAt
E U” [ o
=
D, E are p x p nonsingular
so E = I.
From
B*B = U” matrix.
0 o 1 U, matrices.
the hypothesis,
so
where F is a p x p nonsingular
= R(AAT).
U, Then
More-
R(AA*)
=
112
IRVING
AA'f'B*BA=U" Similarly
JACK
KATZ
[t :],:,:*[:,~]ZJA=B*BA
one proves
BfBAA"B*=AA*B*. From a theorem
of Greville
[3, Theorem
11, the reverse order law follows.
REFERENCES 1 T. S. Bask&t
and I. J. Katz,
2 E. Bores,, An.
Univ.
x2777. 3 T. N. E. Greville, 4 H. L. Hamburger Vector
Space,
Timisoara SIAM
Penrose,
6 H.
Schwerdtfeger,
Proc.
7 H. Schwerdtfeger, June
Rev.
Introduction
1969;
Sot.
to Linear
Linenr Algebra version
(2) 2(1969),
8(1965) [Math. Reviews
Transfovmations
Press, London
Philos.
Groningen,
revised
and Appl.
87-103. 37(1969j,
518-521.
8(1966),
University
Cambridge
2nd ed., P. Noordhoff,
Received
Algebra
and M. E. Grimshaw, Linear
Cambridge
5 R.
Linear
Ser. St;., Mat.-Fix.
51(1955), Algebra
received
and
1(1968), October
1956.
406-413. the
Theory
1962. and A$@.
in n-Dimensional
and New York,
325-328. 1969
of Matrices,