Internat. J. Hath. & Math. S.
VOL. 15 NO. 2 (1992) 261-266
261
ON POLYNOMIAL EPr MATRICES
AR. MEENAKSHI and N. ANANOAM
Department of Mathematics,
Annamalai Universlty,
Annamalainagar- 608 002,
Tamll Nadu, INDIA.
(Received Hay 8, 1989)
ABSTRACT.
This paper gives a characterization of EP-X-matrices. Necessary and
r
sufficient conditions are determined for {i} the Moore-Penrose inverse of an EP
r
matrix to be an EP-X-matrix and {ii} Moore-Penrose inverse of the product of
r
to be an EP-X-matrix.
EP-X-matrlces
r
Further, a condition for the generalized
r
inverse of the product of X-matrices to be a X-matrix is determined.
KEY WORDS AND PHRASES: EP-X-matrlces, generalized inverse of a matrix.
r
AMS SUBJECT CLASSIFICATION CODES" 15A57, 15A09.
-
1. INTRODUCTION
Let
be the set of all mxn matrices whose elements are polyrmmtals in
X over an arbitrary field F with an involutary automorphism a" a
for a e F.
l}
The elements of
are called x-matrlces. For A(X}
e
X
(alj{
Let
are
X}}of
set
elements
whose
all
A (X}
mxn
matrices
be the
rational functions of the form f( X}/g( X} where f( X}, 8(X # 0 are polynomlals in
X. For simplicity, let us denote A(X) by A itself.
F
F
(jt
The rank of A
eF
F,
is defined to be the order of its largest minor that is
AeF
not equal to the zero polynomial {[2]p.259}.
is said to be an unlmodular
X-matrix {or} invertible in
if the determinant of A{X}, that is, det A{X} is
a nonzero constant.
is said to be a regular X-matrix if and only if it is
of rank n {[2]p.259), that is, if and ortly if the kernel of A contains only the
r and
zero element.
is said to be EP over the field .F(X) if rk (A}
r
$
R(A)
R(A
where R(A}and rk (A) denote the range space of A and rank of A
respectively [4]. We have unlmodular X-matrices
regular x-matrices}
E P- X-matrices }.
Throughout this paper, let AeF]. Let 1 be identity element of F. The
+
Moore-Penrose inverse of A, denoted by A is the unique solution of the following
AF
F
.
AeF
set of equations-
AXA=A (1.1); XAX=X (1.2)" (AX)=AX [1.31; (XA} =XA (1.41
A
+
exists and
exists, A is
(or}
{I
A+F
EPr over
if and orfly if rk
F(X}
+
AA
(AA*)
A+A.
rk
For
(A’A)
AeF,
rk (A) [7]. When A+
a generalized inverse
inverse is defined as a solution of the polynomial matrix equation (1.11
and a reflexive generalized inverse or) {1,2} inverse is defined as a solution of
the equations (1.1} and (1.2} and they belong to
The purpose of this
paper is to give a characterization of an
EPX-matrix.
Some results on
r
EP- X-marices having the same range space are obtained,
As an application
r
necessary and sufficient conditions are derived for (AB) + to be an EP-X-matrix
r
whenever A and B are EP-X-matrices.
r
F.
AR. MEENAKSHI AND N. ANANDAM
262
2. CHARACTERIZATION OF AN EP r-h-MATRIX
THEOREM I. AEFn] is EP r over the field F{k) if and only if there exist
r
an nxn unlmodular k-matrix P and a r x r regular k-matrlx E such that
PAP
0"
D
PROOF. By the Smith’s canonical form, A =1
O
where P and
O are
unimodular-k-matrices of order n and D is a rxr regular diagonal h-matrix. Any
p-1
(1)
where R2, R3, and R
{1} inverse of A is given by A
4
are arbitrary conformable matrices over F(k). A is EP over the field F(k)
Q_1
D-11
R3
----@ R(A)= R(A
AA*(1) A*
A
=-
D
==>
QP
QP
R
3
R
4
*{
R*
1.0 0.,
%
0
(By Theorem 1713])
2
,-I
O
,
O
0
0
-1
Partitioning conformably, let, QP
T3
o o
4
DT
DT2
o
o
T2
0
1
T4,
DO
E
P
DT
1
Conversely, let
T1
0
p,
oo
P
T1
0
P*
P
o
is a r x r regular h-matrix.
E
0
0
0
PAP
where E is a r x r regular h-matrix.
Since E is regular, E is EP
If
0
ITI 0!
QP
A
where
0
0
0
T3
Hence
0
01
DT2R2D
+
D
(since D is regular).
,-1
Therefore
4
T
LT3
=-
T
=@
R(E)
b
R(PAP
==
R(A)
=
Ar]
r
over F(X).
R(E
R(PA P
R(A
over F(X).
A is EP
Hence the theorem
r
EP over the field F{k} then we can find nxn regular
and is
and K need not be unlmodular h-matrlces.
For example, consider A
Ik{=
.A is
263
POLYNOMIAL EP r MATRICES
E P,
being a regular
K
A
-1 *
The
’,XI
Here H
A
there
then
inverse
then
X-matrices.
are not
EPr-X-matrix
H and K
for
condition
nxn unimodular
exist
AK
1
necessary
a
gives
unimodular X-matrices.
If A is an nxn
THEOREM 2.
{1}
ll/k’
and K
A"
If
:0
0
theorem
following
-I
A* A
HA then H
If A
X-matrix.
’-il
-1/X
and A has a
X-matrices H and
to
be
X-matrix
K
such that
AK.
HA
an nxn
PROOF. Let A be an nxn EP r-X-matrix. By Theorem 1, there exists
0
E
where E is a rxr regular
unimodular X-matrix P such that PAP
0
0
-I is also a X-matrix.
inverse, E
Since A has a X-matrix {1
A
x-matrix.
0
I0
0
p-1
[ E*
-I
*
P
P
A
Therefore
E
p-1
A
Now
0
0
E*E-I
p-I
0
I
0
HA where
p-I
H
X-matrix. Similarly we can write A
E-1E
,
K
p-1
P
1
-
pp-I
p-l*
E
0
0
0
o
P is
an nxn
AK where
is an nxn unimodular X-matrix.
0
AK.
HA
A
REMARK I. The converse of Theorem 2 need not bo true.
Therefore
,Ix
consider
,
0
Since A
A, H
0
1} inverse.
X-matrix
A
10
unimodular
K
A is an
12
For example,
EP1- X-matrix.
However A has no
3. MOORE-PENROSE INVERSE OF AN EP -X-MATRIX
r
The following theorem gives a set of necesssry and sufficient conditions for
the existence of the X-matrix Moore-Penrose inverse of a given X-matrix.
THEOREM 3. For A F], the following statements are equivalent.
rk(A
and A A has a X-matrix {1 inverse.
i) A is EP r, rk(A)
2)
ti) There exists an unimodular X-matrix U with
A
U
U
0
where D is a rxr unimodular X-matrix and U U is a diagonal block matrix.
iii) A
GLG where L and G G are rxr unimodular X-matrices and G is a X-matrix.
+
is a X-matrix and EP r
v) There exists a symmetric idempotent
iv) A
,
AE
EA and R(A)
PROOF. (i)
rk(A2),
rk(A)
a X-matrix {1}
pp
(ii)
/
A
(E
2
,
E
E
such that
R(E).
Since
A is an EP
over the field F(X
r X-matrix
and
exists, by Theorem 2.3 of [5]. By Theorem 4 in [6], A A has
inverse
where
0
X-matrix E,
implies
P1
is
that there exists an unimodular X-matrix P with
a symmetric
rxr unlmodular
X-matrix
such that
AR. MEENAKSHI AND N. ANANDAM
264
PA
where W is a rxn,
ol
AA
is a
AA+A
A
o
Pl
PAA+P *
k-matrix and
[0
A(AA+).
Hence by Theorem 2 in [6],
X-matrix of rank r.
Since A is EP
0
!Wo
p-1
A
Therefore
p-1
r’P1AA
A+A
+
,.o
and
0
P
o
L" oJ
0
thus H is a nxr, X-matrix of rank r.
H consists of the first r columns of P
p-I
Now A
rxr regular
D
0
0
O
p-I
U
U
Since A
X-matrix.
and D
WH is a
inverse and P is an
-ID
inverse. Therefore by Theorem I in [6], D
X-matrix
a
has
0
{I}
{I}
X-matrix
hasa
PAA P *
unimodular X-matrix,
p-I
where U
0
is an unimodular X-matrix
D
Hence A
U
where D
which implies D is an unimodular X-matrix.
U
0 0
is a rxr unimodular X-matrix and U U is a diagonal block X-matrix.
Thus (ii) holds.
O]
,
=-
(i)
(iii)
U2
u4
U1
Let us partition U as U
u3
u3
0
0
u4
U U is
Since
a
dtasonal
unimodular X-matrices.
(iii) =)
(iv)
Since A
G(G * G)
+
A
+
Now AA
+
A is RP r.
GLG
1
Then
is a rxr X-matrix.
GLG
u
Lu2
are
U
where U
X-matrices.
3
block
X-matrix,
UIU1
G G
and
U3U 3
+
L are rxr
Thus (iit) holds.
L and G G are unimodular X-matrices.
-IL-I (G*G)-IG *.
One can verify that
A+A
GLG G (G G)
(G G) G
G(G G) G
implies that
Since L and G * G are untmodular, L -1 and (G * G) -1 are X-matrices, and
G is a X-matrix. Therefore A + is a X-matrix. Thus (iv) holds.
(Iv)
-
(v)
=
Proof is analogous to that of (II)
{v)
=
{i)
(lil) of Theorem 2.3 [5].
Since E is a symmetric Idempotent X-matrix with R(A)
Theorem 2.3 in [5] we have A is EP
+
E
Let
e.
AE
A
E
R(E)
and R(A)
and a. denote the
=--
X-matrix,
Aej
aj,
since
==
ej
jth
r+
AA
and rk(A)
EE
columns
+
of
rk(A
E.
E
R(E) and AE
2) ==>
A exists. Since
(AA+)A A.
Now AE
EA
and A respectively.
Then
is a A-matrix, the equation Ax
has a X-matrix solution.
EA, by
+
aj
where
aj
is a
Hence by Theorem 1 in [6] it follows that
+
A ha a X-matrix {I} inverse.
Further AA
E is also a X-matrix.
Hence by
Theorem 4 in [6] we see that A A has a X-matrix {1} inverse. Thus (i) holds.
Hence the theorem.
POLYNOMIAL EP r MATRICES
265
The condition (i) in Theorem 3 cannot be weakened which can be
REMARK 2.
seen by the following examples.
EXAMPLE I.
is EP
and
A
1
X
rk(A}
rk(A
1.
A A
has
X-matrix 1} inverse (since
| 2
Consl,dec
2}
thel 2XmJatrlx2Xzl’A21
,
L2X
the tnvariant polynomial of A A is
A
A,
I
4X
+
1
1
1
1
X-
is not a
EXAMPLE 2. Consider the matrix
I/
which is not the identity of F). For this
X-matrix. Thus the theorem falls
A
rk{A2),
rk(A)
A A"
0
0
.0
{I} inverse (since any conformable X-matrix is a
this A, A
does not exist.
REMARKS 3.
X-matrix,
and
A is EP I. Since
has a X-matrix
over GF(5).
X-matrix {I
inverse}.
For
Thus the theorem fails.
From Theorem 3, it is clear that if E is a symmetric idempotent
A is a
X-matrix such that R{E}
R{A} then A is EP
+
z=>
AE
EA
A is a X-matrix and EP.
We can show that the set of all EP-X-matrices with common range space as
r
that of given symmetric tdempotent X-matrix forms a group, analogous to that of
the Theorem 2.1 in [5].
COROLLARY 1. Let E
E*
e
Then
H(E}={A e
R(E}} is s maximal subgroup of
A is EP over F(X} and R(A}
r
containing E as identity.
E2
F:
F.
F
PROOF.
This can be proved similar to that of Theorem 2.1 of [5]
applying Theorem 3.
4. APPLICATION
by
In general, if A and B are X-matrices, having X-matrix {1} inverses,
it is not
the
X-matrix
1} inverse for both A and B.
tnvariant polynomial of AB is 1+2 X2
But AB
0
3
+2 X
Since
the
0
1, AB has no X-matrix {1 inverse.
The followin 8 theorem leads to the existence of X-matrix {1}
tnverse of the
product AB.
2
THEOREM 4.
Let A, B
If A
A and B has X-matrix {1} inverse
and R(A} .C. R(B} then AB has a X-matrix 1
inverse.
PROOF. Suppose ABx
b, where b ls a X-matrix, is a consistent system.
Then b
R(AB) R{A) c._ R{B) and
therefore Bz
b.
Since ]9 has a
o
X-matrix 1 inverse, by Theorem I in [6] we get z is a X-matrix. Since
A is
o
idempotent, so in particular A is a{1 }inverse of A and b R{A}, we have
Ab=b.
Now
Ab
b.
Thus ABx
b has a X-matrix solution.
Hence by Theorem1
in [6], AB has a X-matrix 1} inverse. Hence the theorem.
The converse of Theorem 4 need not be true which can be seen by
the
following example.
1
0
1
1
1
EXAMPLE 4. Let A
B
AB
Here
0
0
X
X
0
F.
ABZo
266
AR. MEENAKSHI AND N. ANANDAM
R (A)
R(B).
we
Hence the converse is not true
Next
shall
discuss the necessary and sufficient condition for the
Moore-Penrose inverse of the product of EP-k-matrices to be an EP-k-matrix.
THEOREM 5.
Let A and B be EP-k-matrices.
Then A A has a k-matrix
rk(A)
rk(A
and R(A)
R(B) if and only if AB is EP and
2)
{1} inverse,
+
+
(AB)
is a k-matrix
PROOF.
Since
A
and
B
are
EP
with
R(A)
R(B)
and
r
rk(A)
by a Theorem of Katz [1], AB is EP
Since A is a
EP-k-matrix rk(A)
rk(A
and A A has a k-matrix {1 inverse, by Theorem 3
+
A is a k-matrix and there exists a symmetric idempotent k-matrix E such that
B+A
rk(A2),
2)
+
+
Hence AA
AA
E.
Since A and B are EP and R(A)
R(B),
r
+
+
we have AA
BB
E
Therefore BE
EB and R(B)
R(E).
Again from Theorem 3, for the EP -k-matrix B, we see that B + is a k-matrix
r
+
Since A and B are EP
with R(A)
we can verify that (AB) +
R(B)
+
+
+
Since B and A are k-matrices, it follows that (AB) is a k-matrix.
+
+
Conversely, if (AB)
is a
k-matrix and AB is EP
then (AB)
is an
r
EP-k-matrix.
Therefore
by
Theorem
3
there
a
exists
symmetric
idempotent
r
R(A)
R(E).
A+A
B+B.
B+A
_
+
+
k-matrix E such that R(AB)
R(E) and (AB) (AB)
E
(AB)
(AB).
Since rk(AB)
rk(A)
r and R(AB)iR(A), we get R(A)
R(E). Since A is EP
r’
+
by Remark 3, it follows that A is a EP-k-matrix.
Now by Theorem 3, A A has
a k-matrix I} inverse and rk(A}
Since AB and B are EP
R(E)
R(AB)
R((AB)
R(B
R(B) and
rk(AB)
rk(B) implies
R(B)
R(E). Therefore R(A)
R(B). Hence the theorem.
rk{A2)r,
REMARK 4. The condition that both A and B are EP-k-matrices, is essential
r
illustrated as follows"
in Theorem 5, is
Let A
and
B
A
and
B
are
0
A A
has a
not EP
1
(AB)
+
ACKNOWLEDGEMENT
k-matrix
1
I
1} in verse and R(A)
R(B).
not
EP
1.
But AB is
0
is not a k-matrix. Hence the claim
1+4 k
2k 0
The authors wish to thank the referee for suggestions which
greatly improved the proofs of many theorems.
REFERENCES
I.
2.
3.
4.
5.
6.
7.
KATZ
l.J. Theorems on Products of EP Matrices II
Lin Alg. Appl. I0
r
(1975), 37-40.
LANCASTER, P. and TISMENETSKY, M.
Theory of Matrices, 2nd ed.,
Academic Press, 1985
MARSAGLIA, G. and STYAN, G.P.H
Equalities and inequalities for ranks of
matrices, Lin. and Multi. A1R. 2 (1974), 269-292.
MARTIN PEARL, On Normal and EP Matrices, Mich. Math. J. 6 (1959), 1-5.
r
MEENAKSHI, AR. On EP Matrices with Entries from an Arbitrary Field,
Lin. and Multi. A_lg __9 (1980), 159-164.
M EENAKSHI, AR. and ANANDAM, N.
Polynomial Generalized Inverses of
r.
Polynomial Matrices, (submitted).
PEARL, M.H. Generalized Inverses of Matrices with Entries Taken from an
Arbitrary Field, Lin. AIR. Appl. 1 (1968), 571-587.