International Journal of Scientific and Innovative Mathematical Research (IJSIMR)
Volume 2, Issue 12, December 2014, PP 940-943
ISSN 2347-307X (Print) & ISSN 2347-3142 (Online)
www.arcjournals.org
Group Inverses of Con-s-k-EP Matrices
B.K.N. Muthugobal
R. Subash
Guest Lecturer in Mathematics
Bharathidasan University Constituent
College, Nannilam
Assistant Professor in Mathematics,
A.V.C College of Engineering,
Mayiladuthurai.
subash_ru@rediffmail.com
Abstract: In this paper, the existence of the group inverse for con-s-k-EP matrices under certain
condition is derived.
Keywords: AMS classification: 15A09, 15A15, 15A57
1. INTRODUCTION
Let
be the space of nxn complex matrices of order n. let
be the space of all complex
n tuples. For A
. Let , ,
, , , R(A), N(A) and
denote the conjugate,
transpose, conjugate transpose, secondary transpose, conjugate secondary transpose, Moore
Penrose inverse range space, null space and rank of A respectively. A solution X of the
equation AXA = A is called generalized inverse of A and is denoted by
. If A
then the unique solution of the equations A XA =A , XAX = X,
= AX ,
XA
XA
[3] is called the Moore-Penrose inverse of A and is denoted by
. A matrix A is called
Con-sif
A r and N(A) = N( VK) (or) R(A)=R(KV ). Throughout this paper
let “ be the fixed product of disjoint transposition in
permutation matrix .
Let us define the function k (x)= xk 1 , xk 2 ,..., xk
n
= { 1,2,….n} and k be the associated
. A matrix A = (
is s-k-
symmetric if =
for i, j = 1,2,…..n . A matrix A
is said to be Con-sk-EP if it satisfies the condition
=0
= 0 or equivalently N(A) =N( VK). In
addition to that A is con-s-k-EP
is con-EP or AVK is con-EP and A is con-s-kEP
is con-s-k-EPr moreover A is said to be Con-s-k-EPr if A is con-s-k-EP and of rank
r. For further properties of con-s-k-EP matrices one may refer [2].
Theorem 2 (p.163) [1]
Let
A Cn
n.
Then
A
is
EP
A#
A†
when
A# exists.
It is well known that, for an con-EP matrix, group inverse exists and coincides with its MoorePenrose inverse. However, this is not the case for a con-s-k-EP matrix. For example, consider
0 0 1
A
0 0 1
0 1 0
for k=(1,2)(3), the associated permutation matrix
0 0 0
K
1 0 0
and
0 0 1
0 0 1
V
0 1 0 .
1 0 0
©ARC
Page | 940
B.K.N. Muthugobal & R. Subash
0 0 0
A
is con-s-k-EP1 matrix, But
A2
0 0 0
,
( A)
( A2 ) .
0 0 0
#
Therefore by Theorem 2 , group inverse A does not exists for A . Here it is proved that for a
con-s-k-EP matrix A , if the group inverse exists then it is also a con-s-k-EP matrix.
Theorem 2.1.1
Let
A Cn
n
be con-s-k-EP r and
( A2 ) .
( A)
Then
A# exists and is con-s-k-EPr.
Proof
( A2 ) , by Theorem 2 , A# exists for A .
#
# T
is enough to prove that R( A ) R( KV ( A ) ).
Since,
Since,
( A)
AA#
To show that
A# is con-s-k-EPr, it
R( AA# )
A# A , we have, R( A)
R( A# A)
R( A# )
AA# A A AT AT ( A# )T AT
KVAT KVAT A#T AT
Therefore,
R( KVAT ) R( KVAT A#T AT )
R( KVAT A#T )
R( KV ( A# A)T )
R( KV ( AA# )T )
R( KVA#T AT )
R( KVA#T )
Now,
A
is
R( AT ) R( KV ( A# )T )
con-s-k-EPr
and
( A)
R( A) R( KVAT ) and
A# is con-s-k-EPr.
( A# ) r
( A) r
Remark 2.1.2
In the above Theorem the condition that
Example 2.1.3
( A)
( A2 )
is essential.
0 0 1
Let
A
0 0 0
for k = (1,2)(3), the associated permutation matrix
0 0 0
0 1 0
K
1 0 0
0 0 1
and V
0 0 1
0 0 0
0 1 0 . KVA
1 0 0
0 0 0
is EP1
A
is
0 0 1
International Journal of Scientific and Innovative Mathematical Research (IJSIMR)
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Group Inverses of Con-s-k-EP Matrices
0 0 0
con-s-k-EP1. Since A
2
0 0 0
( A2 ) 0.
,
0 0 0
( A2 ). Hence A# does not exists for a con-s-k-EP
#
con-s-k-EP matrix A , if A exists then it is also con-s-k-EPr.
That is,
( A)
matrix
A. Thus, for a
Theorem 2.1.4
For
A Cn
( KVA)#
A# exists then, A is con-s-k-EP
if
n
A†VK.
Proof
A is con-s-k-EP
( KVA)# ( KVA)†
KVA is con-EP
(by Theorem (2.11) [2] )
(by Theorem 2)
( KVA)#
A†VK
(by Theorem (2.1.12))
Theorem 2.1.5
For
A Cn
n,
A
A†
is con-s-k-EPr
KV
AVK )
(polynomial in KVA ) VK .
(polynomial in
Proof
( KVA)† f ( KVA) for some
†
commutes with ( KVA) .
( KVA)( KVA)† ( KVA)† (KVA)
( KVA)( A†VK ) ( A†VK )(KVA)
KVAA†VK A†VKKVA
KVAA†VK A† A
KVAA† A† AKV
A is con-s-k-EPr
It is clear that if
(ByTheorem(2.11)[2]) Conversely, let
A
scalar polynomial
1
( KVA) s
1
KVA
then
KVAA† A† AKV and
expressed as KV (polynomial in
be con-s-k-EPr, then
KVA† A AA† KV . Now, we will prove that A† can
AVK ) and (polynomial in KVA ) VK .
s 1
( KVA)s
.... q ( KVA) s q 0 be
1 ( KVA)
Then s 0(or ) s 1. For suppose s 2, then
( KVA)†[( KVA) s
f ( x)
....
q
be
Let
the minimal polynomial of
( KVA) s q ] 0
KVA.
.
Hence,
[( KVA)( KVA)† ( KVA)](KVA)s
2
[(KVA)(KVA)† (KVA)](KVA)s 1 ...
1
q
That is,
( KVA)s 1 1 ( KVA)s ....
If s 0, then
q
[( KVA)( KVA)† ( KVA)]( KVA)s
( KVA)s
q 1
0
q 2
0
this is a contradiction.
International Journal of Scientific and Innovative Mathematical Research (IJSIMR)
Page 942
B.K.N. Muthugobal & R. Subash
( KVA)†
A†VK
A†
A
1
( KVA)
1
A 1VK
1
[
1
1
I
I
2
(polynomial in
Thus,
2
2
( KVA)....
( KVA)....
( KVA)....
KVA ) KV
q
A† (polynomial in KVA ) KV .
( KVA)†[ KVA
1
However,
1
2
( KVA)† ( KVA)
A†VK
[
( KVA)
q
q
q
( KVA)q
( KVA)q
1
1
( KVA)q 1 ]KV
s 1, then
( KVA) q 1 ] 0
( KVA) 2....
q
and
( KVA) q
it
follows
that
is a polynomial in
A
( KVA)† [( KVA)† ( KVA)]( KVA)†
1
A†
If
( KVA)2 ....
( KVA)† ( KVA)
.
I
1
A† AKV
1
A† A
1
A†VKKVA
2
2
2
( KVA)....
2
( KVA)....
( KVA) KV ....
( KVA)....
q
q
q
( KVA) q 1 ]
q
( KVA)q
1
( KVA)q 1 KV
( KVA)q 1 ]KV
KVA ) KV .
(polynomial in KVA ) KV .
(polynomial in
Thus,
†
A
REFERENCES
[1] Ben-Israel, A. and Grevile, T.N.E., Generalized Inverses: Theory and Applications, New
York: Wiley and Sons (1974).
[2] Krishnamoorthy, S., Gunasekaran, K. and Muthugobal, B.K.N., “Con-s-k-EP matrices”,
Journal of Mathematical Sciences and Engineering Applications , Vol. 5, No.1, 2011, 353
– 364.
[3] Rao, C.R. and Mitra, S.K., “Generalized Inverse of Matrices and Its Applications”, Wiley and
Sons, New York, 1971.
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