The characterization of g-inverses, minimum norm g-inverses and least square g-inverses of bimatrices are derived as a generalization of the generalized inverses of matrices.

1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. VI (Jan - Feb. 2015), PP 19-34
www.iosrjournals.org
DOI: 10.9790/5728-11161934 www.iosrjournals.org 19 | Page
On Least Square, Minimum Norm Generalized Inverses of
Bimatrices
G.Ramesh*, P.Maduranthaki**
*Associate Professor of Mathematics, Govt. Arts College(Autonomous), Kumbakonam.
**Assistant Professor of Mathematics, Arasu Engineering College, Kumbakonam.
Abstract: The characterization of g-inverses, minimum norm g-inverses and least square g-inverses of
bimatrices are derived as a generalization of the generalized inverses of matrices.
Keywords: g-inverse, minimum norm g-inverse, least square g-inverse.
AMS Classification: 15A09, 15A15, 15A57.
I. Introduction
Let n nC  be the space of n n complex matrices of order n. A matrix 1 2BA A A  is called a bimatrix if
1A and 2A are matrices of same (or) different orders [7,8] and a bimatrix is called unitary if
* *
B B B B BA A A A I  [8]. For 1 2, n nA A C  , 1 2BA A A  and let  *
,B BA r A and BA
denote the
conjugate transpose, rank and generalized inverse of the bimatrix BA where BA
is the solution of the equation
B B B BA X A A [1,4,6]. In general, the generalized inverse BA
of BA is not uniquely determined. Since BA
is not unique, the set of generalized inverses of BA is some times denoted as  BA
.
In this paper we analyze the characterization of g-inverse, minimum norm g-inverse and least square
g-inverses of bimatrices as a generalization of the g-inverses of matrices [2,3,5] .
II. Generalized Inverses of Bimatrices
In this section some of the properties of generalized inverses of matrices found in [1,2,3,5] are extended
to generalized inverses of bimatrices .
Theorem: 2.1
Let B B BH A A
 and B B BF A A
 . Then the following relations hold :
(i)
2
B BH H and
2
B BF F .
(ii) ( ) ( ) ( )B B Br H r F r A 
(iii)    B Br A r A

(iv)    B B B Br A A A r A 
 .
Proof of (i)
Now
2
.B B BH H H
  B B B BA A A A 

       1 2 1 2 1 2 1 2A A A A A A A A
 
    
       1 2 1 2 1 2 1 2A A A A A A A A   
    
  1 1 2 2 1 1 2 2A A A A A A A A   
  
   1 1 1 1 2 2 2 2A A A A A A A A   
 
  1 1 1 2 2 2 1 2A A A A A A A A   
  

2. On Least Square, Minimum Norm Generalized Inverses of Bimatrices
DOI: 10.9790/5728-11161934 www.iosrjournals.org 20 | Page
     1 2 1 2 1 2 1 2A A A A A A A A   
      
      1 2 1 2 1 2 1 2A A A A A A A A
 
     
  
 B B B BA A A A 

B BA A

.BH
Hence ,
2
B BH H .
Now
2
.B B BF F F
  B B B BA A A A 

       1 2 1 2 1 2 1 2A A A A A A A A   
    
  1 1 2 2 1 1 2 2A A A A A A A A   
  
     1 1 1 1 2 2 2 2A A A A A A A A   
 
     1 2 1 2 1 2 1 2A A A A A A A A   
      
 B B B BA A A A 

B BA A

= BF .
Hence,
2
.B BF F
Proof of (ii)
Now    B B Br A r A A

   B Br A r H (1)
Also  Br A  B B Br A A A

    1 2 1 2 1 2r A A A A A A 
   
    1 1 1 2 2 2r A A A A A A 
 
   1 1 2 2 1 2r A A A A A A 
  
    1 2 1 2 1 2r A A A A A A 
   
  B B Br A A A

 B Br H A
   B Br A r H (2)
From (1) and (2), we get    B Br A r H (3)
Now    B B Br A r A A

   B Br A r F (4)
Also    B B B Br A r A A A

    1 2 1 2 1 2r A A A A A A 
   

3. On Least Square, Minimum Norm Generalized Inverses of Bimatrices
DOI: 10.9790/5728-11161934 www.iosrjournals.org 21 | Page
    1 1 1 2 2 2r A A A A A A 
 
   1 2 1 1 2 2r A A A A A A 
  
    1 2 1 2 1 2r A A A A A A 
   
  B B Br A A A

 B Br A F
   B Br A r F (5)
From (4) and (5), we get    B Br A r F (6)
From (3) and (6) we get      B B Br A r H r F 
.
(7)
Proof of (iii)
Now    B B B Br A r A A A

    1 2 1 2 1 2r A A A A A A 
   
 1 1 1 2 2 2r A A A A A A 
 
 1 1 2 2r A A A A 
 
 1 2r A A 
 
 Br A

Hence,    B Br A r A
 .
Proof of (iv)
Now    B B B B Br A A A r A A 

Also we have    B B B B B B Br A A A r A A A A   

     1 2 1 2 1 2 1 2r A A A A A A A A   
    
    1 1 1 1 2 2 2 2r A A A A A A A A   
 
   1 2 1 1 1 2 2 2r A A A A A A A A   
  
      1 2 1 2 1 2 1 2r A A A A A A A A        
 
 B B B Br A A A A 
   
 B Br A A

 Br H
 Br A ( by (7) )
Hence,    B B B Br A A A r A 
 .
Theorem: 2.2
Let BA be a bimatrix, then the following relations hold for a generalized inverse BA
of BA .
(i)      B BA A
  


4. On Least Square, Minimum Norm Generalized Inverses of Bimatrices
DOI: 10.9790/5728-11161934 www.iosrjournals.org 22 | Page
(ii)  B B B B B BA A A A A A
 

(iii)     
*
* * * *
B B B B B B B BA A A A A A A A
 

Proof of (i)
Let B B B BA A A A

 B B B BA A A A
 

      *
1 2 1 2 1 2 1 2A A A A A A A A

 
    
 
** *
1 2 1 1 1 2 2 2A A A A A A A A 
  
   
* *
1 1 1 2 2 2A A A A A A 
 
     ** * * *
1 1 1 2 2 2A A A A A A
 
 
       
* ** * * * * *
1 2 1 2 1 2 1 2A A A A A A A A 
    
 
** * *
B B B BA A A A

Thus      * *
B BA A

 (8)
On the other hand,  * * *
B B B BA A A A

 and so     
*
*
B BA A
 

     **
B BA A
 
 (9)
From (8) and (9), we get      * *
B BA A

 .
Proof of (ii)
Let   B B B B B B BG A I A A A A
 
 
 B B B B B BA A A A A A
 
 
         1 2 1 2 1 2 1 2 1 2 1 2A A A A A A A A A A A A

   
         
         1 2 1 2 1 2 1 2 1 2 1 2BG A A A A A A A A A A A A
    
       
         1 2 1 2 1 2 1 2 1 2 1 2BG A A A A A A A A A A A A
            
  
           1 2 1 2 1 2 1 2 1 2 1 2A A A A A A A A A A A A

              
  
        1 2 1 1 1 1 1 2 2 2 2 2A A A A A A A A A A A A

          
  
     1 1 1 1 1 1 2 2 2 2 2 2A A A A A A A A A A A A

          
  
     1 1 1 1 1 1 2 2 2 2 2 2A A A A A A A A A A A A
       
   

5. On Least Square, Minimum Norm Generalized Inverses of Bimatrices
DOI: 10.9790/5728-11161934 www.iosrjournals.org 23 | Page
     1 1 1 1 1 1 2 2 2 2 2 2A A A A A A A A A A A A
            
      
   
     1 1 1 1 1 1 2 2 2 2 2 2A A A A A A A A A A A A
          
   
           1 2 1 2 1 2 1 2 1 2 1 2A A A A A A A A A A A A
          
       
           1 2 1 2 1 2 1 2 1 2 1 2BG A A A A A A A A A A A A
    
       
           
         
1 2 1 2 1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2 1 2 1 2
B BG G I I A A A A A A A A A A
A A A A A A A A A A A A
    
    
          
       
  
         
               
1 2 1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
B BG G I I A A A A A A A A
A A A A A A A A A A A A A A A A
  
    
         
           
     B B B B B B B B B B B B B B BG G I A A A A A A A A A A A A
         
  
     B B B B B B B B B B B B B B BG G I A A A A A A A A A A A A
       
  
Let B B BB A A
 .
  B B B B B B B B BG G I B B B B B B  
  
  B B B B BI B B B B
  
0
Hence, 0BG 
That is,    0B B B B B BA I A A A A
 
 
  0B B B B B BA A A A A A
 
 
Hence,  B B B B B BA A A A A A
 
 .
Proof of (iii)
Let BG denote a generalized inverse of B BA A
.Then BG
is also a generalized inverse of B BA A
and
2
B B
B
G G
S


 is a symmetric generalized inverse of B BA A
.
Let  B B B B B B B BH A S A A A A A
  
 
          1 2 1 2 1 2 1 2 1 2 1 2 1 2BH A A S S A A A A A A A A A A

     
           
          1 2 1 2 1 2 1 2 1 2 1 2 1 2BH A A S S A A A A A A A A A A
                
  
             1 2 1 2 1 2 1 2 1 2 1 2 1 2BH A A S S A A A A A A A A A A

                   
  

6. On Least Square, Minimum Norm Generalized Inverses of Bimatrices
DOI: 10.9790/5728-11161934 www.iosrjournals.org 24 | Page
      1 1 1 2 2 2 1 1 1 1 2 2 2 2A S A A S A A A A A A A A A

            
  
     1 1 1 1 1 1 1 2 2 2 2 2 2 2A S A A A A A A S A A A A A

            
  
     
* *
1 1 1 1 1 1 1 2 2 2 2 2 2 2A S A A A A A A S A A A A A
        
   
         1 1 1 1 1 1 1 2 2 2 2 2 2 2A S A A A A A A S A A A A A
              
      
   
     * *
1 1 1 1 1 1 1 2 2 2 2 2 2 2AS A A A A A A S A A A A A         
   
      * *
1 1 1 2 2 2 1 1 1 1 2 2 2 2AS A A S A A A A A A A A A         
   
            
* *
1 2 1 2 1 2 1 2 1 2 1 2 1 2A A S S A A A A A A A A A A
                     
            * *
1 2 1 2 1 2 1 2 1 2 1 2 1 2BH A A S S A A A A A A A A A A
                       
            
            
* *
1 2 1 2 1 2 1 2 1 2 1 2
* *
1 2 1 2 1 2 1 2 1 2 1 2 1 2
B BH H A A S S A A A A A A A A
A A S S A A A A A A A A A A
       
      
        
  
        
  
          
              
* *
1 2 1 2 1 2 1 2 1 2
* *
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
A A S S A A A A A A
A A A A S S A A A A A A A A A A A A
    
          
       
  
          
  
   * * * * * *
B B B B B B B B B B B B B BA S A A A A A S A A A A A A
        
      
   * * * * * * *
B B B B B B B B B B B B B B B BH H A S A A A A A S A A A A A A
      
      
Let  *
B B BS A A


 * * * * *
B B B B B B B B B B B B B BH H A S A S A A S A A A S A
    
*
0B BH H  .
Hence , 0BH  .
That is ,  * * *
0B B B B B B BA S A A A A A

  .
     
*
* * * *
0B B B B B B B BA A A A A A A A
 
 
    
*
* * * *
B B B B B B B BA A A A A A A A
 
 .

7. On Least Square, Minimum Norm Generalized Inverses of Bimatrices
DOI: 10.9790/5728-11161934 www.iosrjournals.org 25 | Page
III. Minimum norm generalized inverses of bimatrices
In this section some of the characteristics of minimum norm g-inverses of matrices found in [2,3] are
extended to minimum norm g-inverses of bimatrices.
Definition: 3.1
A generalized inverse BA
that satisfies both B B B BA A A A
 and  B B B BA A A A
 
 is called a
minimum norm g-inverse of BA and is denoted by  B m
A

.
Theorem: 3.2
Let BA be a bimatrix, then the following three conditions are equivalent:
(i)     
*
B B B Bm m
A A A A
 
 and  B B B Bm
A A A A


(ii)   * *
B B B Bm
A A A A


(iii)    * *
B B B B B Bm
A A A A A A


.
Proof of (i) (ii)
From (i),  B B B Bm
A A A A


  
*
*
B B B Bm
A A A A


    
*
1 2 1 2 1 2m
A A A A A A

    
 
       
*
1 2 1 2 1 2m m
A A A A A A
 
    
 
   
*
1 1 1 2 2 2m m
A A A A A A
 
  
 
     
* *
1 1 1 2 2 2m m
A A A A A A
 
 
     
* *
* * * *
1 1 1 2 2 2m m
A A A A A A
 
 
       
* *
* * * *
1 2 1 2 1 2( )m m
A A A A A A
          
  
*
* *
B B Bm
A A A
    
  
*
*
B B Bm
A A A


  *
B B Bm
A A A


Hence,   * *
B B B Bm
A A A A

 .
Proof of (ii) (iii)
From (ii),  * *
B B B Bm
A A A A


Postmultiply by  *
B B BA A A

on both sides
     * * * *
B B B B B B B B B Bm
A A A A A A A A A A
 

            * * * *
1 2 1 2 1 2 1 2 1 2 1 2m m
A A A A A A A A A A A A
 
      
       * * * *
1 1 1 2 2 2 1 1 2 2 1 2m m
A A A A A A A A A A A A
 
   

8. On Least Square, Minimum Norm Generalized Inverses of Bimatrices
DOI: 10.9790/5728-11161934 www.iosrjournals.org 26 | Page
          * * * *
1 1 1 2 2 2 1 1 2 2 1 2m m
A A A A A A A A A A A A
  
   
          * * * *
1 1 1 2 2 2 1 1 2 2 1 2m m
A A A A A A A A A A A A
   
   
            * * * *
1 1 2 2 1 2 1 2 1 2 1 2m m
A A A A A A A A A A A A
    
     
           
* ** *
1 1 2 2 1 1 2 2 1 2 1 2m m
A A A A A A A A A A A A
     
    
          
* *
1 1 2 2 1 1 2 2 1 1 2 2m m
A A A A A A A A A A A A
     
   
        
*
1 2 1 2 1 1 2 2B Bm m
A A A A A A A A A A
    
   
   B B B B B Bm
A A A A A A
  

 B B B Bm
A A A A
 
 ( by definition 3.1)
( )B m BA A
 ( by definition 3.1)
Hence,    * *
B B B B B Bm
A A A A A A

 .
Proof of (iii) (i)
From (ii) of theorem (2.2),  * *
B B B B B BA A A A A A


Replace BA by
*
BA we get
    * * * * * *
B B B B B BA A A A A A
 

 * * * *
B B B B B BA A A A A A


        * * * * * * *
1 2 1 2 1 2 1 2 1 2BA A A A A A A A A A A

     
       * * * * * *
1 2 1 2 1 2 1 2 1 2A A A A A A A A A A
 
     
          * * * * * *
1 2 1 2 1 2 1 2 1 2A A A A A A A A A A
         
  
     * * * * * *
1 1 1 1 2 2 2 2 1 2A A A A A A A A A A
  
  
     * * * * * *
1 1 1 1 2 2 2 2 1 2A A A A A A A A A A
 
  
         * * * * * *
1 2 1 1 2 2 1 2 1 2A A A A A A A A A A
 
     
  
 * * *
B B B B BA A A A A

 
  
 * *
B B B Bm
A A A A

 ( by (iii))
    * *
B B B Bm
A A A A
 

      * *
1 2 1 2 1 2B m m
A A A A A A A

 
    
 
     * *
1 1 1 2 2 2m m
A A A A A A

 
  
 

9. On Least Square, Minimum Norm Generalized Inverses of Bimatrices
DOI: 10.9790/5728-11161934 www.iosrjournals.org 27 | Page
     
*
* *
1 1 1 2 2 2m m
A A A A A A
 
 
     
* *
* *
1 1 1 2 2 2m m
A A A A A A
 
 
        * *
1 2 1 2 1 2m m
A A A A A A
       
 
  
*
*
B B B B m
A A A A


Premultiply by  * *
B B BA A A

on both sides
       
*
* * * * *
B B B B B B B B B B m
A A A A A A A A A A
  

       
*
*
B B B B B Bm m m
A A A A A A
  
 ( by (iii))
  
*
*
B B m
A A

 ( by(ii))
  
*
B Bm
A A


Hence,     
*
B B B Bm m
A A A A
 
 .
Also, from (iii) of theorem (2.2),
    
*
* * * *
B B B B B B B BA A A A A A A A
 

Replace BA by
*
BA we get
         
* * * ** * * * * * * *
B B B B B B B BA A A A A A A A
 
 
  
 
    
*
* * * *
B B B B B B B BA A A A A A A A
 

         
*
* * * *
1 2 1 2 1 2 1 2B Bm
A A A A A A A A A A
      
  
    
*
* * * *
1 2 1 1 2 2 1 2A A A A A A A A

    
  
       
*
* * * *
1 2 1 1 2 2 1 2A A A A A A A A
 
    
  
       
*
* * * *
1 2 1 1 2 2 1 2A A A A A A A A
      
  
     
*
* * * *
1 1 1 1 2 2 2 2A A A A A A A A
    
  
     
* *
* * * *
1 1 1 1 2 2 2 2A A A A A A A A
   
  
 
         
* ** ** * * *
1 1 1 1 2 2 2 2A A A A A A A A
     
    
   
       * ** *
1 1 1 1 2 2 2 2A A A A A A A A   
 

10. On Least Square, Minimum Norm Generalized Inverses of Bimatrices
DOI: 10.9790/5728-11161934 www.iosrjournals.org 28 | Page
        
* ** *
1 2 1 2 1 2 1 2A A A A A A A A   
    
 
**
B B B BA A A A 

 
*
B B B BA A A A 

 B B B BA A A A 
 ( by definition 3.1)
 B B B Bm
A A A A
 
 ( by definition 3.1)
Premultiply by BA on both sides,
 B B B B B Bm
A A A A A A
 

 B B B Bm
A A A A

 ( by definition 3.1)
Hence,  B B B Bm
A A A A

 .
Theorem: 3.3
Let BA be a bimatrix, then the following relations hold for a minimum norm generalized inverse
 B m
A

of BA :
(i) One choice of  *
B B m
A A

is     
*
.B Bm m
A A
 
(ii) One choice of    1
B Bm m
A A 
 
 where  is a non- zero scalar.
(iii) One choice of  B B B m
U A V

is  B B Bm
V A U
 
where BU and BV are unitary bimatrices.
(iv)     
*
* *
B B B B B B mm
A A A A A A
 

.
Proof of (i)
Now   * * *
1 2 1 2B BA A A A A A  
* *
1 1 2 2A A A A 
   * * *
1 1 2 2B B m m
A A A A A A
 
 
   * *
1 1 2 2m m
A A A A
 
 
       * *
1 1 2 2m mm m
A A A A
  
 
         * *
1 2 1 2m mm m
A A A A
   
  
       
* *
1 2 1 2m m m
A A A A
      
 
      
*
1 2 Bm m m
A A A
  
 
    
*
B Bm m
A A
 
 .
Hence,       
*
*
B B B Bm mm
A A A A
  
 .

11. On Least Square, Minimum Norm Generalized Inverses of Bimatrices
DOI: 10.9790/5728-11161934 www.iosrjournals.org 29 | Page
Proof of (ii)
Now    1 2B m m
A A A 

   
 1 2 m
A A 

 
   1 2m m
A A 
 
 
   1 1
1 2m m
A A 
  
 
    1
1 2m m
A A
 
 
 1
B m
A


Hence ,    1
B Bm m
A A 
 
 .
Proof of (iii)
      1 2 1 2 1 2B B B m m
U A V U U A A V V

   
 1 1 1 2 2 2 m
U AV U AV

 
   1 1 1 2 2 2m m
U AV U AV
 
 
           1 1 1 2 2 2m m m m m m
V A U V A U
     
 
             * * * * * * * *
1 1 1 1 1 1 1 2 2 2 2 2 2 2m m
V VV A U U U V V V A U U U
    
 
since    * *
B B B Bm
A A A A


             * * * *
1 1 1 1 1 2 2 2 2 2m m
V I A U I V I A U I
     
 
( Since 1 2 1 2, , &U U V V are unitary matrices )
     * * * *
1 1 1 2 2 2m m
V A U V A U
 
 
       * * * *
1 2 1 2 1 2m m
V V A A U U
 
   
 * *
B B Bm
V A U

 .
Hence,    * *
B B B B B Bm m
U A V V A U
 
 .
Proof of (iv)
        * * * * * *
1 2 1 2 1 2 1 2B B B Bm m
A A A A A A A A A A A A

    
   * * * *
1 1 2 2 1 1 2 2m
A A A A A A A A

  
     * * * *
1 1 2 2 1 1 2 2m m
A A A A A A A A
 
  
         * * * *
1 1 1 1 2 2 2 2m mm m
A A A A A A A A
  
 
          
* *
* *
1 1 1 1 2 2 2 2m m m m
A A A A A A A A
      
 
          
* *
* *
1 2 1 1 1 2 2 2m m m m
A A A A A A A A
       
 

12. On Least Square, Minimum Norm Generalized Inverses of Bimatrices
DOI: 10.9790/5728-11161934 www.iosrjournals.org 30 | Page
            
* *
* *
1 2 1 2 1 2 1 2m m m m
A A A A A A A A
            
     
*
*
B B B Bm m
A A A A
 

  
*
*
B Bm
A A

 ( by (ii) of theorem 3.2)
  
*
B B m
A A


Hence,     
*
* *
B B B B B B mm
A A A A A A
 
 .
IV. Least Square Generalized Inverses of Bimatrices
In this section some of the characteristics of least square g-inverses of matrices found in [2,3,5] are
extended to least square g-inverses of bimatrices .
Definition: 4.1
A generalized inverse BA
that satisfies both B B B BA A A A
 and 
*
B B B BA A A A 
 is called a least
square generalized inverse bimatrix of BA and is denoted by B l
A

.
Theorem: 4.2
Let BA be a bimatrix, then the following three conditions are equivalent:
(i)  B B B Bl
A A A A

 and     B B B Bl l
A A A A
 

(ii)  * *
B B B Bl
A A A A


(iii)    * *
B B B B B Bl
A A A A A A


.
Proof of (i) (ii)
From (i),  B B B Bl
A A A A


  
*
*
B B B Bl
A A A A


         *
1 2 1 2 1 2B l l
A A A A A A A

 
   
    
*
1 1 1 2 2 2l l
A A A A A A
 
 
     1 1 1 2 2 2l l
A A A A A A
  
 
     
* *
* * * *
1 1 1 2 2 2l l
A A A A A A
 
 
         
* *
* * * *
1 2 1 2 1 2l l
A A A A A A
          
  
*
* *
B B Bl
A A A
   
 
  
*
*
B B B l
A A A


 *
B B B l
A A A



13. On Least Square, Minimum Norm Generalized Inverses of Bimatrices
DOI: 10.9790/5728-11161934 www.iosrjournals.org 31 | Page
Hence ,  * *
B B B Bl
A A A A

 .
Proof of (ii) (iii)
From (ii),  * *
B B B Bl
A A A A


Premultiply by  *
B B BA A A

on both sides
     * * * *
B B B B B B B B B B l
A A A A A A A A A A
  

             * * * *
1 2 1 2 1 2 1 2 1 2 1 2l l
A A A A A A A A A A A A
  
      
            * * * *
1 2 1 2 1 2 1 2 1 2 1 2l l
A A A A A A A A A A A A
  
      
              * * * *
1 2 1 2 1 2 1 2 1 2 1 2l l
A A A A A A A A A A A A
    
      
         * * * *
1 1 1 1 1 1 2 2 2 2 2 2l l
A A A A A A A A A A A A
   
 
         * ** *
1 1 1 1 1 1 2 2 2 2 2 2l l
A A A A A A A A A A A A
    
 ( by (i) theorem 2.2)
              * * * *
1 2 1 2 1 2 1 2 1 2 1 2l l
A A A A A A A A A A A A
    
      
   *
B B B B B B l
A A A A A A
  

   B B B B B B l
A A A A A A
  

  B B B B B B l
A A A A A A
 
 ( by definition 4.1)
  B B B B l
A A A A

 ( by definition 4.1)
 B B l
A A

 ( by definition 4.1)
Hence,    * *
B B B B B Bl
A A A A A A

 .
Proof of (iii) (i)
From (iii) of theorem (2.2),
    
*
* * * *
B B B B B B B BA A A A A A A A
 

          
*
* * * *
1 2 1 2 1 2 1 2B B l
A A A A A A A A A A

     ( by (iii))
      
*
* * * *
1 2 1 2 1 2 1 2A A A A A A A A
     
  
        
*
* * * *
1 2 1 2 1 2 1 2A A A A A A A A
       
  
     * * * *
1 1 1 1 2 2 2 2A A A A A A A A

    
  
     * * * *
1 1 1 1 2 2 2 2A A A A A A A A
   
 

14. On Least Square, Minimum Norm Generalized Inverses of Bimatrices
DOI: 10.9790/5728-11161934 www.iosrjournals.org 32 | Page
         
* ** ** * * *
1 1 1 1 2 2 2 2A A A A A A A A
     
    
   
         
* ** * * ** *
1 1 1 1 2 2 2 2A A A A A A A A      
    
   
     * ** *
1 1 1 1 2 2 2 2A A A A A A A A   
 
        
* * * *
1 2 1 2 1 2 1 2A A A A A A A A   
    
  *
B B B BA A A A
 

 
*
B B B BA A A A 

 B B B BA A A A 
 ( by definition 4.1)
 B B B Bl
A A A A
 
 ( by definition 4.1)
Postmultiply by BA on both sides
 B B B B B Bl
A A A A A A
 

 B B B Bl
A A A A

 ( by definition 4.1)
Hence ,  B B B Bl
A A A A

 .
Also, from (ii) of theorem (2.4),
 * *
B B B B B BA A A A A A


  
*
* * *
B B B B B BA A A A A A


        
*
* * * *
1 2 1 2 1 2 1 2 1 2A A A A A A A A A A

      
  
       
*
* * * *
1 2 1 2 1 2 1 2 1 2A A A A A A A A A A
      
  
         
*
* * * *
1 2 1 2 1 2 1 2 1 2A A A A A A A A A A
        
  
     
* *
* * * *
1 1 1 1 1 2 2 2 2 2A A A A A A A A A A
   
  
 
         
* ** ** * * * * *
1 1 1 1 1 2 2 2 2 2A A A A A A A A A A
  
 
     * ** * * *
1 1 1 1 1 2 2 2 2 2A A A A A A A A A A   
 
      * * * * * *
1 1 1 1 1 2 2 2 2 2A A A A A A A A A A
  
 
     * * * * * *
1 1 1 1 1 2 2 2 2 2A A A A A A A A A A
 
 
         * * * * * *
1 2 1 2 1 1 2 2 1 2A A A A A A A A A A
 
     
  
   * * * *
1 2 B B B BA A A A A A

  
  

15. On Least Square, Minimum Norm Generalized Inverses of Bimatrices
DOI: 10.9790/5728-11161934 www.iosrjournals.org 33 | Page
 * *
B B B B l
A A A A

 ( by (iii))
    
*** *
B B B B l
A A A A


      
*** * * *
1 2 1 2 1 2 1 2 l
A A A A A A A A

    
       
*
* *
1 2 1 2 1 2 1 2l l
A A A A A A A A
 
     
 
     
*
* *
1 1 1 2 2 2B l l
A A A A A A A
 
  
 
     
* *
* *
1 1 1 2 2 2l l
A A A A A A
 
 
     
* *
* *
1 1 1 2 2 2l l
A A A A A A
 
 
        
* *
* *
1 2 1 2 1 2l l
A A A A A A
      
 
  
*
*
B B B Bl
A A A A


Postmultiply by  * *
B B BA A A

on both sides
      
*
* * * * *
B B B B B B B B B Bl
A A A A A A A A A A
   
  
      
*
*
B B B B B Bl l l
A A A A A A
  
 ( by (iii) )
  
*
*
B Bl
A A

 ( by (ii) )
  
*
B B l
A A


Hence,   
*
B B l
A A

=  B B l
A A

.
Theorem: 4.3
Let BA be a bimatrix, then the following relations hold for a least square generalized inverse  B l
A

of
BA :
(i)    1
B Bl l
A A 
 
 where  is a non zero scalar.
(ii) One choice of    * *
B B B B B Bl l
U A V V A U
 
 where BU and BV are unitary bimatrices.
Proof of (i)
Now    1 2B l l
A A A 

   
 1 2 l
A A 

   
   1 2l l
A A 
 
 
   1 1
1 2l l
A A 
  
 
    1
1 2l l
A A
 
 
 1
B l
A


Hence,    1
B Bl l
A A 
 
 .

16. On Least Square, Minimum Norm Generalized Inverses of Bimatrices
DOI: 10.9790/5728-11161934 www.iosrjournals.org 34 | Page
Proof of (ii)
Now       1 2 1 2 1 2B B B l l
U A V U U A A V V

   
    1 1 1 2 2 2 l
U AV U AV

 
   1 1 1 2 2 2l l
U AV U AV
 
 
             1 1 1 2 2 2l l l l l l
V A U V A U
     
 
             * * * * * * * *
1 1 1 1 1 1 1 2 2 2 2 2 2 2l l
V V V A U U U V V V A U U U
    
 
    * *
since B B B Bl
A A A A


             * * * *
1 1 1 1 1 2 2 2 2 2l l
I V A I U I V A I U
     
 
(since 1 2 1 2, ,U U V and V are unitary bimatrices)
     * * * *
1 1 1 2 2 2l l
V A U V A U
 
 
       * * * *
1 2 1 2 1 2l l
V V A A U U
 
   
 * *
B B Bl
V A U


Hence,    * *
B B B B B Bl l
U A V V A U
 
 .
References
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