On Products of Polynomial Conjugate EPr Matrices

On Products of Polynomial Conjugate EPr Matrices

On Products of Polynomial Conjugate EPr Matrices

S. Krishnamoorthy

Head of the Department of Mathematics, Ramanujam Research Centre, Department of Mathematics,

Government Arts College(Autonomous), Kumbakonam-612 001, Tamil nadu(India).

G. Manikandan, Lecturer in Mathematics,

Ramanujam Research Centre, Department of Mathematics, Government Arts College(Autonomous),

Kumbakonam-612 001, Tamil nadu(India).

Abstract

In this paper we disscuss the product of polynomial

inverse of A satisfying the following four equations:(1) AXA A, (2) XAX X ,

conjugate EPr (con- EPr ) matrices is polynomial

(3) AX* AX , (4) XA* XA

[2].

A* is the

con- EPr .

Keywords: EP matrix, polynomial matrix,

conjugate transpose of A . In general product of two polynomial con- EPr matrices need not be

i 0

Generalized inverse.

AMS classification:15A09, 15A15, 15A57.

polynomial con- EPr . For instance,

0 0

0 0

and

0 0

0 i

are polynomial con- EP1

matrices, but the

1. Introduction

   Throughout this paper we deal with complex polynomial square matrices. An nn square matrix A() which is a polynomial in the

   scalar variable from a field C represented by A() Amm Am1m1 ……. A1 A0 where the leading coefficient Am 0 , Ai, s are square

   product is not polynomial con- EP1 matrix. The

   purpose of this paper is to answer the question of when the product of polynomial con- EPr matrices is

   polynomial con- EPr , analogous to that of EPr matrices studied by [1]. We shall make use of the following results on range space, rank and generalized inverse of a matrix.

   matrices in

   Vnn

   is defined a polynomial matrix.

   (1) R(A) R(B) AA BB

   Any matrix A is said to be polynomial con- EPr if

   (2)

   R(A ) R(A*)

   R(A) R(AT ) or equivalently N(A) N(AT ) or

   (3)

   (A) (A )

   equivalently AA AA and is said to be

   (AT ) (A)

   polynomial con- EPr if A is polynomial con- EPr and

   (4)

   (A ) A .

   (A) r , where R(A), N(A), A,AT and (A) denote

   the range space, null space, conjugate, transpose and

   rank of A respectively. A denotes the Moore- Penrose

2. ON PRODUCTS OF POLYNOMIAL CONJUGATE EPr MATRICES

   In this section, Explain the product of polynomial

   conjugate EPr (con- EPr ) matrices is polynomial con- EPr

   Theorem 2.1

   A is polynomial con- EPr . Hence the theorem.

   Corollary 2.2

   Let A and B be polynomial con- EPr matrices.

   Then AB is a polynomial con- EPr matrices

   (AB) r and R(A) R(B) .

   Let

   A1 and

   An (n 1)

   be polynomial con- EPr

   Proof:

   matrices and let A A1A2A3……An . Then the following statements are equivalent.

   1. A is polynomial con- EPr .

   2. R(A1) R(An ) and (A) r

      1 n

      1 n

   3. R(A*) R(A* ) and (A) r

      Proof follows from Theorem 1 for the product of two matrices A, B .

      Remark 2.3

      In the above corollary both the conditions that

      (AB) r and R(A) R(B) are essential for a

   4. A polynomial con- EPr .

Proof:

n

n

1. (ii): Since R(A) R(A1) and rk(A) rk(A1) we get R(A) R(A1) . Similarly R(AT ) R(AT ) .

   product of two polynomial con- EPr matrices to be

   polynomial con- EPr . This can be seen in the following:

   Example 2.4

   Now, A is polynomial con- EPr R(A) R(AT )

   Let

   A 1 i ,

   B i 1

   be polynomial

   i 1 1 i

   and (A) r (by definition of polynomial con- EPr )

   n

   n

   R(A1) R(AT ) and (A) r

   R(A1) R(An ) and (A) r

   (since An is polynomial con- EPr )

2. (iii)

con- EPr matrices. Here R(A) R(B) , (AB) 1

and AB is not polynomial con- EP1 .

Example 2.5

R(A ) R(A ) A A A A

(by result (1))

Let A i 0 and

B i i by

1 n 1 1 n n

0 0

i i

A A A A

1 1 n n

polynomial con- EPr matrices. Here R(A) R(B) ,

AA1 A An (since A1 , An

1 n

are polynomial con- EPr )

(AB) 1 and AB is not polynomial con- EP1 .

R(A ) R(A ) (by results

1 n

1 n

1 n

(1)and (4))

Remark 2.6

In particular for A B, corollary 1 reduces to the

Therefore,

R(A*) R(A* )

(by results (2))

following.

1 n

1 n

R(A1) R(An ) and (A) r R(A*) R(A* )

and (A) r .

1. (i):

   A is polynomial con- EPr R(A ) R(A )T and

   (A ) r (by definition of polynomial con- EPr )

   R(A) R(A) and (A ) r

   R(AT ) R(A) and (A) r

   Corollary 2.7

   Let A be polynomial con- EPr . Then Ak is polynomial con- EPr (Ak ) r .

   Theorem 2.8

   Let (AB) (B) r1 and (BA) (A) r2 . If

   AB , B are con EPr1 and A is con EPr2 , then

   (by results (2)and (3))

   BA is con EPr2 .

   Proof

   Since

   (BA) (A) r2 ,it is enough to show that

   Remark 2.11

   For any two polynomial con- EPr matrices

   N(BA) N(BA)T . N(A) N(BA) and (BA) (A)

   A and B , since AB , AB ,

   A B ,

   A B , A B ,

   implies N(BA) N(A) . Similarly N(AB) N(B) .

   BA

   all have the same rank, the property of a

   Now,

   N(BA) N(A)

   N(AT )

   (Since A is polynomial

   matrix being polynomial con- EPr is preserved for its conjugate and Moore-Penrose inverse, by applying Corollary 1 for a pair of polynomial con- EPr matrices

   r

   r

   con- EP )

   2

   among A , B , A , B , A , B , A , B and using

   r

   r

   con- EP )

   1

   N(BTAT )

   N((AB)T )

   N(AB)

   (Since AB is polynomial

   the result 2, we can deduce the following.

   Corollary 2.12

   Let A , B be polynomial con- EPr matrices.

   Then the following statements are equivalent.

   1. AB is polynomial con- EPr matrices.

      N(B) (Since N(AB) N(B) )

      con- EPr )

      N(BT ) (Since B is polynomial

   2. AB is polynomial con- EPr matrices.

   3. A B is polynomial con- EPr matrices.

      N(ATBT ) N(BAT ) .

   4. A B

      is polynomial con- EPr matrices

      Further, (BA) (BA)T implies N(BA) N(BA)T . Hence the Theorem.

A B is polynomial con- EPr matrices

BA is polynomial con- EPr matrices

Lemma 2.9

If A , B are polynomial con- EPr matrices and

AB has rank r , then BA has rank r .

Proof:

(AB) (B) dimN(A) N(B* ) .

Theorem 2.13

If A , B are polynomial con- EPr matrices.

R(A) R(B) then (AB) BA .

Proof:

Since A is polynomial con- EPr and R(A) R(B) ,

Since (AB) (B) r , N(A) N(B*) 0

we have

R(A ) R(B) . That is given

x Cn

(the

N(A) N(B*) 0 N(A) N(B) 0

(Since

set of all n1

complex matrices) there exists a

B is polynomial con- EPr )

N(A) N(B) 0

N(A*) N(B) 0

(Since

yCn such that Bx A y . Now,

Bx A y BAABx BAAA y BA y

BBx

A is polynomial con- EPr ).Now,

Since BB is hermitian, it follows that

BAAB is

*

hermitian. Similarly, R(A ) R(B) implies

(BA) (A) dimN(B) N(A ) r 0 r

ABBA is hermitian. Further by result (1),

Hence the Lemma.

Theorem 2.10

If A , B and AB AB are polynomial con- EPr

matrices, then BA is polynomial con- EPr .

Proof:

Since A , B are polynomial con- EPr matrices and

(AB) r , by Lemma 1, (AB) r . Now the result follows from Theorem 2, for r1 r2 r .

AA BB . Hence,

AB(BA )AB ABB(BB) B

AB

(BA)AB(BA) B(BB) BBA

BA

Thus BA satisfies the defining equations of the Moore-Penrose inverse, that is, (AB) BA . Hence the theorem.

Remark 2.14

In the above Theorem, the condition that

Now, set C Ir 0

K I

and consider

R(A) R(B) is essential.

nr

Ir 0 E EL I KT

CACT r

Example 2.15

K Inr KE KEL 0 Inr

Let

A i i and

B i 0

Here A and

E EL I

KT E EKT EL

i i

0 0

r

0 0

0 I 0 0

B are polynomial con- EP1 matrices, (AB) 1 ,

nr

R(A) R(B) and (AB) BA

Remark 2.16

The converse of Theorem 4, need not be true in general. For,

CACT is polynomial con- EPr From

N(A) N(CACT ) it follows that EL EKT 0 , and so L KT ,completing the proof.

Theorem 2.18

Let

A i 0

0 0

0 0

and

B 0 0 . A and B are

0 i

0 i

If A , B are polynomial con- EPr matrices,

(AB) r and (AB) BA , then R(A) R(B) .

polynomial con- EPr matrices, such that

(AB) BA , but R(A) R(B) .

Next to establish the validity of the converse of the Theorem 4, under certain condition, first let us prove a Lemma.

Proof:

Since A is polynomial con- EPr , by Theorem 3 in [3], there is a unitary matrix U such that,

0 0

0 0

UTAU D 0 , where D is rr nonsingular

Lemma 2.17

E F

matrix.

Set U*BU B1

B3

B2

B4

Let

A G H be an nn polynomial con-

UT ABU UTAUU*BU D 0 B1

B2

EPr matrix where E is an rr matrix and if E F

0 0 B B

has rank r then E is nonsingular. Moreover there is

3 4

DB1 DB2

E EKT

0 0

an (n r) r matrix K such that A .

1 2

1 2

KE KEKT

D 0 B B

Proof:

has

r

r

I 0

0 Inr 0 0

Since A is polynomial con- EPr , is

rankr and thus.

0 0

B B D 0

polynomial con- EPr and E Fhas rank r , the U*BAU U*BUUTAU 1 2

r

r

I 0 E F

E F

B3

B4 0 0

product

is a product of

B1D 0

0 0 G H 0 0

B D 0

polynomial con- EPr matrices which has rank r .

3

Therefore by Lemma 1 the product

B1 0 D 0

has

B3 0 0 I

E F Ir 0 E 0

has rank r . Hence there

n r

G H 0 0 G 0

B1 B2 B1 0

rankr . It follows that 0 0 and B 0 have

is an (n r) r matrix K and an r(n r) matrix L such that G KE , F EL , and E is nonsingular.

KE KEL

KE KEL

Therefore, A E EL

3

rankr , so that B1 is nonsingular.

* B1

B1KT

Theorem 2.19

By Lemma 2,

U BU , with

KB KB KT

Let A , B are polynomial con- EPr matrices,

1 1

(AB) r and (AB) BA , then AB is

(U*BU) (B1 ) r . By using Penrose representation for the generalized inverse [4], we get

polynomial con- EPr .

Proof:

B*PB* B*PB*K* T

(U*BU) 1 1 1 1

where

R(B) R(B ) (Since B is polynomial con- EPr )

KB*PB* KB*PB*K* *

1

1

1

1

1

1

1

1

1 1 1 1

P (B1 B* B1 KT KB*)1 B1(B*B1 B*K* KB1)1

R(B) R(B )

R(B*A*)

(Since R(B*A*) R(B*)

UTBU (U*BU)

Q QK*

KQ KQK*

where

and (AB)* (AB) r (B*)

R(AB)* R(AB)

(by result (2))

Q (I KT K)1B1(I K*K)1

R(AB )

(by hypothesis)

* T

1

1

D1 0

R(A ) R(A*) R(A) (by result (2)

U A U (U AU)

0 0 and A is polynomial con- EPr ).

UTABU UTABU(UTABU) UTABU

UTABU(UT (AB) U)UTABU

is unitary)

UTABU(UTB A U)UTABU

(byhypothesis)

(since U

R(B) R(A) R(B) R(A)

Since (AB) r R(B) R(A) , by Corollary 1, AB

is polynomial con- EPr . Hence the Theorem.

UTABU(UTB U)(U*A U)UTABU

(since U is unitary). On simplification, we get,

DB1QB1 DB2 KQB1 DB1

DB1(I B1 B2 K)QB1 DB1

1

1

Since B2 B1KT , QB1 (I KT K)1 . Hence (I KT K) (QB1)1 I . Thus KT K 0 which implies K*K 0 so that K 0 .

0 0

0 0

U*BU B1 0

UTAU D 0

0 0

U* A U D 0

0 0

0 0

Since D and B1 are rr nonsingular matrices we have

References:

1. T.S. Baskett and I.J.Katz, Theorems on Products of EPr matrices, Linear Algebra Appl., 2(1969), 87-103.

2. A. Ben Israel and T.N.E.Greville, Generalized inverses- Theory and Applications, Wiley, Interscience, New York, 1974.

3. AR. Meenakshi and R. Indira, On conjugate EP matrices periodica math., Hung.,

4. R. Penrose, On best approximate solutions of linear matrix equations, Proc.Cambridge Phil.

   Soc., 52(1956), 17-19

5. C.R. Rao and S.K. Mitra, Generalized inverse of matrices and its applications Wiley and Sons, New York, (1971).

D 0 B

1

1

R(D) R(B ) R R

0

0 0

0 0

0 0

0 0

1

R(U* A U) R(U*BU)

R(A) R(B).

Hence the Theorem.