On Real Matrices to Least-Squares g-Inverse and Minimum Norm g-Inverse of Quaternion Matrices

(1)

On Real Matrices to Least-Squares g-Inverse and

Minimum Norm g-Inverse of Quaternion Matrices

*

Huasheng Zhang

School of Mathematics Science, Liaocheng University, Shandong, China Email: zhsh0510@yahoo.com.cn

Received November 20, 2011; revised December 24, 2011; accepted December 30, 2011

ABSTRACT

Through the real representations of quaternion matrices and matrix rank method, we give the expression of the real ma-trices in least-squares g-inverse and minimum norm g-inverse. From these formulas, we derive the extreme ranks of the real matrices. As applications, we establish necessary and sufficient conditions for some special least-squares g-inverse and minimum norm g-inverse.

Keywords: Extreme Rank; g-Inverse; Least-Squares g-Inverse; Minimum Norm g-Inverse; Quaternion Matrix

1. Introduction

Throughout this paper, stands for the real number field, stands for the set of all m matrices over the quaternion algebra



m n

 n





2 2 2 0 1 2 3

0 1 2 3

= = =

, , , .

a a i a j a k i j k ijk a a a a

   



 = = 1,

I, AT, A* and _A†

stand for the identity matrix’ the transpose’ the conjugate transpose and the Moore-Pen- rose inverse of a quaternion matrix A. In [1], for a qua-ternion matrix A, dim

 

A = dim

 

A . dim

 

A

†

is called the rank of a quaternion matrix A and denoted by r A

 

.

The well known Moore-Penrose inverse A of

m n

A  is defined to be the unique matrix _X_m n

satisfying the following four Penrose equations 1) AXA=A,

2) XAX =X, 3)

 

AX = AX,

 

= .



4) XA XA

A matrix X is called a least-squares g-inverse of A if it satisfies both 1) and 3) in the Penrose equations, and de-noted by a matrix X is called a minimum norm g-inverse of A if it satisfies both 1) and 4) in the Penrose equations, and denoted by The general expres-sion of

 1,3

;

A

 1,3

 1,4

.

A

A and A 1,4 can be written as

 1,3 †

= A ,

A A L V (1)

 1,4 †

= A

A A WR (2)

where , the two matrices V

and W is a arbitrary; see [[2], pp. 44-46].

†

= ,

A

L IA A RA =IAA†

For convenience of representation, we suppose

0 1 2 3

=

A A A iA jA k (3) and

   

1,3

0 1 2 3

1,4

0 1 2 3

= ,

=

A B B i B j B k

A C C i C j C k

  

   (4)

where 0, 1, 2, 3 , m n

A A A A   B B B B₀, ₁, ₂, ₃m n ,

0, 1, 2, 3 . n m

C C C C 

For an arbitrary quaternion matrix

1 2 3 4

= ,

M M M iM jM k

we define a map ( ) from m n to _4m4n by

 

4 3 2 1

3 4 1 2

2 1 4 3

1 2 3 4

= .

M M M M

M

M M M M



 

_ _ 

 

   

 

 

 

(5)

By (5), it is easy to verify that ( ) satisfies the follow-ing properties:

a) M =N 

 

M =

 

N .

b) 



MN



=

   

M  N , 



MN



=

   

M  N ,

 

kM =k

 

M ,k

  _.

c)

 

1

 

1



= m n= m



n

M T M T R M R

 _  _ 

 

=Sm M Sn,

1 

*

This research was supported by the Natural Science Foundation of China

(2)

0 0 0 0 0 0

= , = ,

0 0 0 0 0 0

0 0 0

= , = , .

0 0 0

t t

t

t t

t

I I

T R

I I

S t m n

I I

 

  

   _

  



  



 

 

  

 

 

     

d) r_

 

M _= 4r M

 

.

The least-squares g-inverse and minimum norm g-inverse have a very wide range of applications in nu-merical analysis and mathematical statistics and have been examined by many authors(see, e.g., [3-6]). Haruo [3] developed some equivalent conditions on least-squares general inverse in 1990. Tian [4] presented the maximal and minimal ranks of the Schur complement to least-squares g-inverse and minimum norm g-inverse in 2004. Tian [5] establish necessary and sufficient conditions for a matrix to be the least-squares g-inverse and minimum norm g-inverse from rank formulas in 2005. Guo, Wei and Wang [6] derived structures of least squares g-inverses and minimum norm g-inverse of a bordered matrix in 2006.

Quaternion matrices play an important role in me-chanics, computer science, quantum physics, signal and color image processing and so on. More and more inter-ests of quaternion matrices have been witnessed recently (e.g. [7-14]).

Noticing that the properties of the real matrices in least-squares general inverse A 1,3 and minimum norm g-inverse A 1,4 (4) have not been considered so far in the literature. We in this paper use the real representa-tions of quaternion matrices and matrix rank method to investigate (4) over . In Section 2, we first give the expression of the real matrices i and

in (4), then determine the maximal and minimal ranks of the real matrices i and i in (4). As

applications, we establish necessary and sufficient condi-tions for a quaternion matrix has a pure real or pure imaginary



B

= 0

C i





= 0,1, 2, 3

i C i



2, 3



,1,

1,3

A and A1,4. The necessary and suffi-cient conditions for all A1,3 and A 1,4 are pure real or pure imaginary of a quaternion matrix are also presented.

2. Main Results

We begin with the following lemmas which proof just like those over the complex field.

Lemma 2.1 (see[15]) Let A, B and C be matrices over . Then

, ,

m n m k l n   

a) r A B



,



=r A

  

r R BA



=r B

  

r R AB



,

L

b) =

  

A



=

 



C



A

r r A r CL r C r AL

C

 

 

   

c) =

    



.

0 B C

A B

r r B r C r R A

C

 

 

 

 

Lemma 2.2 (see [16]) Let _A_m n ,

,

m p

B 

q n

C be given. Then

a) The maximal rank of ABXC with respect to X is









max = min , ;

X

A

r A BXC r A B r

C

  

 

 _  _

 

  (6)

b) The minimal rank of ABXC with respect to X is









min = .

0

X

A A B

r A BXC r A B r r

C C

   

  _{ } _ _

    (7)

Theorem 2.3 Suppose _X_ij_



i j, = 1, 2,3, 4



is a least-

4 4

squares g-inverse of 

 

A , where X_ijn m , A and

 1,3

A are defined as (3) and (4). Then in (4) can be written as



1, 2, 3



= 0,

i B i

















0 11 22 33 44

1 12 21 43 34

2 13 31 24 4

3 41 14 32 2

1

= ,

4 1

= ,

4 1

= ,

4 1

= .

4

B X X X X

  

  

  

  

2

3

(8)

Written in an explicit form, in (4) are





= 0,1, 2, 3

i B i

 

       

† †

0 1 1 2 2

† †

3 3 4 4

1 2

1 2 3 4

3 4

1 1

=

4 4

1 1

4 4

_A, _A, _A, _A ,

B P A Q P A Q

P A Q P A Q

V V P L P L P L P L

V V

   

 



 

     

 

    

   

(9)

 

       

† †

1 1 2 2 1

† †

4 3 3 4

2 1

1 2 4 3

3 4

1 1

=

4 4

1 1

4 4

_A, _A, _A, _A ,

B P A Q P A Q

P A Q P A Q

V V P L P L P L P L

V V

   

 



 

  _   

 

    

_   

(3)

 

       

† †

2 1 3 2 4

† †

3 1 4 2

3

1

1 3 2 4

4 2 1 1 = 4 4 1 1 4 4

A , A , A , A ,

B P A Q P A Q

P A Q P A Q

V V

P L P L P L P L

V V                    _   _{ }        (11)

 

        † †

3 1 4 2 3

† †

3 2 4 1

4

1

1 4 3 2

2 3 1 1 = 4 4 1 1 4 4 1 , , , ,

4 A A A A

B P A Q P A Q

P A Q P A Q

V V

P L P L P L P L

V V                             (12) where















1 2 3 4

= , 0, 0, 0 , = 0, , 0, 0 , = 0, 0, , 0 , = 0, 0, 0, ,

n n

P I P I

















1 2

3 4

= , 0, 0, 0 , = 0, , 0, 0 , = 0, 0, , 0 , = 0, 0, 0, ,

T T

m m

T T

m m

Q I Q I

and V1, V2, V3 and V4 are arbitrary real matrices with

compatible sizes.

Proof. Suppose





4 4 , = 1, 2, 3, 4 ij

X i j



   

 

,

is a least- squares g-inverse of  A where X n m, i.e.

ij 

 





 

4 4

4 4 4 4

= ,

= .

ij

ij ij

A X A A

A X A X

                    

Then applying property (c) of ( ) above to them yields

 



 





 

1 1 1

4 4

1 1

4 4 4 4

= ,

= ;

m n ij m n m

m n ij m n ij

T A T X T A T T A T

T A T X T A T X

                         



n 

 



 





 

1 1 1

4 4

1 1

4 4 4 4

= ,

= ;

m n ij m n m n

m n ij m n ij

R A R X R A R R A R

R A R X R A R X

                         





 



 





 

1 1 1

4 4

1 1

4 4 4 4

= ,

= .

m n ij m n m n

m n ij m n ij

S A S X S A S S A S

S A S X S A S X

                        



 Hence,

 





 

1 4 4 1 1

4 4 4 4

= ,

= ;

n ij m

n ij m n ij m

A T X T A A

A T X T A T X T

                       

 





 

1 4 4 1 1

4 4 4 4

= ,

= ;

n ij m

n ij m n ij m

A R X R A A

A R X R A R X R

                       

 





 

1 4 4 1 1

4 4 4 4

= ,

n ij m

n ij m n ij m

A S X S A A

A S X S A S X S

                       

which implies that 1 1 and

4 4 , 4 4

n ij m n ij m

T X T R X R

 

   

   

1 4 4

n ij m

S X S



 

  are also least-squares g-inverses of

 

A

 . Thus,





1 1

4 4 4 4 4 4

1 4 4

1 4

ij n ij m n ij m

n ij m

X T X T R X R

S X S

                         

is also a least-squares g-inverse of 

 

A , where



1 1

4 4 4 4 4 4

1

4 4 = 4 4

ij n ij m n ij m

ij

n ij m

X T X T R X R

S X S X

                               and          

11 ₁₁ ₂₂ ₃₃ ₄₄ 12 ₁₂ ₂₁ ₄₃ ₃₄ 13 ₁₃ ₃₁ ₂₄ ₄₂ 14 ₄₁ ₁₄ ₃₂ ₂₃ 21 ₂₁ ₁₂ ₃₄ ₄₃ 22 ₁₁ ₂₂ ₃₃ ₄₄ 23 ₄₁ ₁₄ ₃₂ ₂₃ 24 ₂₄ ₄₂ ₁₃ ₃₁ 31 ₁₃ ₃₁ ₂₄ ₄₂ 32 ₄₁ ₁₄ ₃₂ ₂₃

= , = , = , = , = , = , = , = , = , = ,

X X X X X

                                   

33 ₃₃ ₄₄ ₁₁ ₂₂ 34 ₂₁ ₁₂ ₃₄ ₄₃ 41 ₄₁ ₁₄ ₃₂ ₂₃ 42 ₂₄ ₄₂ ₁₃ ₃₁ 43 ₁₂ ₂₁ ₄₃ ₃₄ 44 ₃₃ ₄₄ ₁₁ ₂₂

= , = , = , = , = , = .

X X X X X

(4)

Let



_

_













11 22 33 44 12 21 43 34 13 31 24 42 41 14 32 23

1 =

4 1

,

4

X X X X X

X X X X i

X X X X j

X X X X k

  

   

   

   

Then, by (5),



 





1

4 4 4 4

1 1

4 4 4 4

1 =

4

,

ij n ij m

n ij m n ij m

X X T X T

R X R S X S

 

 

 

 

    

   

   

 _ _  _ _

is a least-squares g-inverse of 

 

A . Hence, by the property (a) of ( ) we know that Xˆ a least-squares g-inverse of A. The above discussion shows that the least-squares g-inverse of 

 

A and the least-squares g-inverse of A are equivalent. Observe that Xij, i, j =

1,2,3,4 in (8) can be written as

ˆ

= .

ij i j

X P XQ

From (1), the least-squares g-inverse of 

 

A can be written as



 

† _{ }

= _A

X  A L_ V

where V =



V V V V1, 2, 3, 4



and

4 1, 2, 3, 4

p q

V V V V   are

arbitrary.

Substituting them into (8) yields the four real matrices

B0, B1, B2 and B3 in (9)-(12).

According to Lemma 2 and Theorem 2, we can get the following extreme ranks formulas for the real matrices in the least-squares g-inverses.

Theorem 2.4 Suppose that A and A 1,3 are defined as (3) and (4). Then

a) maxr B

 

₀ = min



r A_ 0 A _ 4r A

 

n m,



;

 

₀ 0 

 



minr B =r A_ A _ r A .

b) maxr B

 

₁ = min



r A 1 A  4r A

 

n m,



;

 

₁ 1 

 



minr B =r A_ A _ r A .

c) maxr B

 

₂ = min



r A 2 A  4r A

 

n m,



;

 

₂ 2 

 



minr B =r A_ A _ r A .

d) maxr B

 

₃ = min



r A 3 A  4r A

 

n m,



;

 

₃ 3 

 



minr B =r A_ A _ r A , where

   

 

   

 

0 1 2

1 0 0

0 1 2 3

2 2 3

3 3 1

0

1 2 3

1 ₀ ₃ ₂

0

2 3 0 1

2 1 0

3

= , = , = , =

0 0 0

= , =

0 0 0

A A A

A A A A

A A A

A

3

0

1

2

,

A

A A

A A A

A

A A

A A A

A

A A A

A

_  _  _   

       

       

      

 

       

      

  __ _ _ _

  _ _



  _ _

  __ _

  _ _

  _  _

 

,



,

and

   

 



0 1 2 3

1 0 3 2

0 1 2 3

2 3 0 1

3 2 1 0

0 2 3 0 1 3

1 3 2 1 0 2

1 2

2 0 1 2 3 1

3 1 0 3 2 0

0 1 3

= , = , = , =

= , =

=

A A A A

A A A A A A

A A

A A A A A A

A A A

A



 

       

      _{ }

       

       

      



    

  

   



   

 _   _ 

   

    

 _   

   

  ₂

1 0 3

2 3 0

3 2 1

.

A A A

 

 

  

 



 

Proof. Applying (6) and (7) to B0 in (9), we get the

following

 

  

 

    

0

max = min , ;

min = ,

0

m

m P

r B r P P r

I

P P

r B r P P r P r

I

  

 _ _{  }  _ _{  }

  

 

 

 _  _{ } _     __ __

where



 

 _{ } _{ } _{ } _{ }

† †

1 1 2 2

† †

3 3 4

1 2 3 4

1 1

=

4 4

1 1

,

4 4

= _A , _A, _A, _A

P P A Q P A Q

P A Q P A Q

P P L_ P L_ P L_ P L_

 



  4

.

 

 

(5)

  

 



_{ }



_{ }



_{ }



_{ }

 



_{ }

1 2 3 4

0 1 2 3 0

0 0 0 0

= 0 0 0 0 4

0 0 0 0

0

0 0 0

= 4

0 0 0

= 4 ,

r P P

P P P P P

A

r r A

A A

P P P P

A A

r r

A A

r A A r A n



 



 

 

 

 

 

 _{  }_ __

 

 

 

 

  

   

 

 

   

 

where    A A A A0, 1, 2, 3 and A are defined as above. By

the same manner, we can get extreme ranks of B1, B2 and

B3.

As one of important applications of the maximal and minimal ranks to real matrices, Theorem 2 can help to get the necessary and sufficient conditions for the exis-tence of some special least-squares g-inverses. We show them in the following.

Corollary 2.5 Suppose A= A₀A i₁ A j₂ A k₃ . Then

a) A quaternion matrix A has a real least-squares g-inverse if and only if

  

 



1 = 2 = 3 =

r A_ A_ r A_ A_ r A_ A_ r A .

b) All least-squares g-inverses of quaternion matrix A are real matrices if and only if

  

_{ }

1 = 2 = 3 = 4

r A_ A_ r A_ A_ r A_ A_ r A n,

where A A A1, 2, 3 and A are defined as Theorem 2.

Corollary 2.6 Suppose A= A₀A i₁ A j₂ A k₃ . Then a) A quaternion matrix A has a pure imaginary least- squares g-inverse if and only if



 



0 = .

r A_ A_ r A

b) All least-squares g-inverses of quaternion matrix A are pure imaginary matrices if and only if



_{ }

0 = 4 ,

r A_ A_ r A n where A0 and A are defined as Theorem 2.

The following several theorems of minimum norm g-inverse can be shown by a similar approach, and their proofs are omitted here.

Theorem 2.7 Suppose





is a

4 4 , = 1, 2, 3, 4 ij

Y _ i j

   

minimum norm g-inverse of 

 

A , where

A and

,

n m ij

Y  

 1,4

A are defined as (3) and (4) Then in (4) can be written as





= 0,1, 2,3

i C i

















0 11 22 33 44

34

42

23

1 12 21 43

2 13 31 24

3 41 14 32

1

= ,

4 1

= ,

4 1

= ,

4 1

= .

4

C Y Y Y Y

  

  

  

  

Written in an explicit form, C i_i = 0,1,



2,3



in (4) are

 





       

2

4

1

2

3

4

,

Q

R Q

 

 

 

† †

0 1 1 2

† †

3 3 4

1 2 3 4

1 1

=

4 4

1 1

4 4

1

, , ,

4

A

C P A Q P A

P A Q P A

W W W W

   

 



 

  

 





       

1

3 4

2

1

3

4

,

Q

R Q

 

 

 

† †

1 1 2 2

† †

4 3

1 2 4 3

1 1

=

4 4

1 1

4 4

1

, , ,

4

A

C P A Q P A

P A Q P A

W W W W

   

 



 

  

 





       

4 2

3 1 2 4

, A

A

Q Q

R Q

 

 

 

† †

2 1 3 2

† †

3 1 4

1 3 2 4

1 1

=

4 4

1 1

4 4

1

, , ,

4

C P A Q P A

P A Q P A

W W W W

   

 



 

  

 





       

† †

3 1 4 2

3 2 4

1 4 3 2

1 1

=

4 4

1 1

4 4

1

, , ,

4

A

C P A Q P A

P A Q P

R Q

W W W W

R Q

   

 



 



3

† †

1 4 1 2 3

,

Q A Q

 

 

 

(6)

where



























1 2

3 4

1 2

3 4

= , 0, 0, 0 , = 0, , 0, 0 ,

= 0, 0, , 0 , = 0, 0, 0, ,

= , 0, 0, 0 , = 0, , 0, 0 ,

= 0, 0, , 0 , = 0, 0, 0, ,

n n

T T

m m

T T

m m

P I P I

Q I Q I



and W1, W2, W3 and W4 are arbitrary real matrices with

compatible sizes.

According to Lemma 2 and Theorem 3, we can get the following extreme ranks formulas for the real matrices in minimum norm g-inverse.

Theorem 2.8 Suppose that A and A 1,4 are defined as (3) and (4). Then

a)

 

0 0

max = min 4 , 4 ;

A

r C r r A m n

A

   

   

 _{ }_ _ 

 

 

 

 

   

 

 

 



 



0 0

minr C =r A r A .

A

        

b)

 



 

1 1

maxr C = min r A 4r A m n, 4 ;

A

  

    

      

  

 



 



1 1

minr C =r A r A .

A

        

c)

 



 

2 2

maxr C = min r A 4r A m n, 4 ;

A

   

 _{  } _ 

 

    

 

 



 



2 2

minr C =r A r A .

A

        

d)

 



 

3 3

maxr C = min r A 4r A m, 4n ;

A

   

 _{  } _ 

 

    

 

 



 



3 3

minr C =r A r A ,

A

        

where

















0 ₀ ₁ ₂ ₃

1 ₁ ₃ ₃ ₂

2 ₂ ₃ ₀ ₁

3 ₃ ₂ ₁ ₀

=

= ,

 



0

1

2

3

2 3 0 1

0 ₁ ₀ ₃ ₂

0 1 2 3

3 2 1 0

1 ₁ ₀ ₃ ₂

0 1 2 3

3 2 1 0

2 ₂ ₃ ₀ ₁

0 1 2 3

3 2 1 0

2 ₂ ₃ ₀ ₁

1 0 3 2

0 0 0

= ,

0 0 0

= ,

=

A

A A

A

A A A A

A A A A A

A A A A

A A A A A

A A A A

A A A A A

A A A A

A A A A A

A A A A

 

 

 

 

 

 _ _ 

 

   

 

 

 _ _ 

 

   

 

 

_ _ 

 

   

 



 



.



 

 _

As one of important applications of the maximal and minimal ranks to real matrices, Theorem 4 can help to get the necessary and sufficient conditions for the exis-tence of some special minimum norm g-inverse. We show them in the following.

Corollary 2.9 Suppose A=A₀A i₁ A j₂ A k₃ . Then

a) A quaternion matrix A has a real minimum norm g-inverse if and only if

  

 



1 2 3

= = =

A A A

r r r r

A A A

     

     

     

.

A

b) All minimum norm g-inverse of quaternion matrix A are real matrices if and only if

  

 

1 2 3

= = = 4

A A A

r r r r A

A A A

     

      

     

     

,

m

where A A A1, 2, 3 and A are defined as Theorem 4.

Corollary 2.10 Suppose A= A₀A i₁ A j₂ A k₃ . Then

a) A quaternion matrix A has a pure imaginary mini-mum norm g-inverse if and only if



 



0

= .

A

r r

A

       

A

,

A A A A A

  



b) All minimum norm g-inverse of quaternion matrix A are pure imaginary matrices if and only if



 

0

= 4 ,

A

r r A

A

 

  

   

m

(7)

REFERENCES

[1] T. W. Hungerford, “Algebra,” Spring-Verlag Inc., New York, 1980.

[2] A. Ben-Israel and T. N. E. Greville, “Generalized Inver- ses: Theory and Applications,” R. E. Krieger Pubilshing Company, New York, 1980.

[3] H. Yanai, “Some Generalized Forms a Least Squares g-Inverse, Minimum Norm g-Inverse, and Moore-Penrose Inverse Matrices,” Computational Statistics & Data Ana- lysis, Vol. 10, No. 3, 1990, pp. 251-260.

doi:10.1016/0167-9473(90)90005-3

[4] Y. Tian, “More on Maximal and Minimal Ranks of Schur Complements with Applications,” Applied Mathematics and Computation, Vol. 152, No. 3, 2004, pp. 675-692. doi:10.1016/S0096-3003(03)00585-X

[5] Y. Tian, “Schur Complements and Banachiewicz-Schur Forms,” Electronic Journal of Linear Algebra Society, Vol. 13, 2005, pp. 405-418.

[6] W. Guo, M. Wei and M. Wang, “On Least Squares g-In- verses and Minimum Norm g-Inverses of a Bordered Ma-trix,” Linear Algebra and its Applications, Vol. 416, No. 2-3, 2006, pp. 627-642.doi:10.1016/j.laa.2005.12.015

[7] S. L. Adler, “Quaternionic Quantum Mechanics and Quan- tum Fields,” Oxford University Press, Oxford, 1995. [8] F. Zhang, “Quaternions and Matrices of Quaternions,”

Linear Algebra and its Applications, Vol. 251, No. 15, 1997, pp. 21-57.doi:10.1016/0024-3795(95)00543-9

[9] C. E. Moxey, S. J. Sangwine and T. A. Ell, “Hypercom-plex Correlation Techniques for Vector Images,” IEEE Transactions on Signal Processing, Vol. 51, No. 7, 2003, pp. 1941-1953.doi:10.1109/TSP.2003.812734

[10] N. LE Bihan and J. Mars, “Singular Value Decomposition of Matrices of Quaternions: A New Tool for Vector-Sensor Signal Processing,” Signal Processing, Vol. 84, No. 7, 2004, pp. 1177-1199.doi:10.1016/j.sigpro.2004.04.001

[11] N. Le Bihan and S. J. Sangwine, “Quaternion Principal Component Analysis of Color Images,” IEEE Interna-tional Conference on Image Processing (ICIP), Barcelona, 14-17 September 2003, pp. 809-812.

[12] N. Le Bihan and S. J. Sangwine, “Color Image Decom-position Using Quaternion Singular Value Decomposi-tion,” IEEE International conference on Visual Informa-tion Engineering (VIE), Guildford, 7-9 July 2003, pp. 113-116.

[13] S. J. Sangwine and N. Le Bihan, “Quaternion Singular Value Decomposition Based on Bidiagonalization to a Real or Complex Matrix Using Quaternion Householder Transformations,” Applied Mathematics and Computation, Vol. 182, No. 1,2006, pp. 727-738.

doi:10.1016/j.amc.2006.04.032

[14] Q. W. Wang, H. S. Zhang and S. W. Yu, “On Solutions to the Quaternion Matrix Equation AXB+CYD=E,” Elec-tronic Journal of Linear Algebra,Vol. 17, 2008, pp. 343- 358.

[15] S. K. Mitra, “A Pair of Simultaneous Linear Matrix Equations and a Matrix Programming Problem,” Linear Algebra and Its Applications, Vol. 131, No. 1, 1990, pp. 97-123.doi:10.1016/0024-3795(90)90377-O