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.
Forest laws, Indian forest laws, why they are important
On Least Square, Minimum Norm Generalized Inverses of Bimatrices
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
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* *
* *
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
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* ** ** * * *
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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