Open Access

Inequalities for ranks of matrix expressions involving generalized inverses

Journal of Inequalities and Applications20142014:87

https://doi.org/10.1186/1029-242X-2014-87

Received: 18 September 2013

Accepted: 10 February 2014

Published: 20 February 2014

Abstract

In this paper, we present several inequalities for ranks of the matrix expressions D A B X A B with respect to the choice of X, where X is taken, respectively, as B ( 1 ) A ( 1 ) , B ( 1 , 2 ) A ( 1 , 2 ) , B ( 1 , 3 ) A ( 1 , 3 ) , B ( 1 , 4 ) A ( 1 , 4 ) , B ( 1 , 2 , 3 ) A ( 1 , 2 , 3 ) as well as B ( 1 , 2 , 4 ) A ( 1 , 2 , 4 ) and B A . Various application of these inequalities are also presented.

MSC:15A09, 15A24.

Keywords

matrix expressiongeneralized inverseinequalityrankgeneralized Schur complement

1 Introduction

Throughout this paper C m × n denotes the set of all m × n matrices over the complex field C. I k denotes the identity matrix of order k, O m × n is the m × n matrix of all zero entries (if no confusion occurs, we will drop the subscript). For a matrix A C m × n , A , R ( A ) , N ( A ) and r ( A ) denote the conjugate transpose, the range space, the null space, and the rank of the matrix A, respectively.

For A C m × n , a generalized inverse X of A is a matrix which satisfies some of the following four Penrose equations [1]:
{ 1 } A X A = A , { 2 } X A X = X , { 3 } ( A X ) = A X , { 4 } ( X A ) = X A .
(1)

For a subset { i , j , , k } of the set { 1 , 2 , 3 , 4 } , the set of n × m matrices satisfying the equations { i } , { j } , , { k } from among equations { 1 } - { 4 } is denoted by A { i , j , , k } . Arbitrary matrix from A { i , j , , k } is called an { i , j , , k } -inverse of A and is denoted by A ( i , j , , k ) . For example, an n × m matrix X of the set A { 1 } is called a g-inverse of A and is denoted by X = A ( 1 ) . The well-known seven common types of generalized inverses of A introduced from (1) are, respectively, the { 1 } -inverse (g-inverse), { 1 , 2 } -inverse (reflexive g-inverse), { 1 , 3 } -inverse (least square g-inverse), { 1 , 4 } -inverse (minimum norm g-inverse), { 1 , 2 , 3 } -inverse, { 1 , 2 , 4 } -inverse and { 1 , 2 , 3 , 4 } -inverse. The unique { 1 , 2 , 3 , 4 } -inverse of A is denoted by A , which is called the Moore-Penrose inverse of A. For convenience, the symbols E A and F A stand for the two orthogonal projectors E A = I m A A and F A = I n A A . We refer the reader to [24] for basic results on generalized inverses.

The notion of rank of a matrix appears to have been introduced [5], by Sylvester in 1851. Two principal classical results [6, 7] on rank are Sylvester’s law of nullity and Frobenius’ inequality. A modern inequality, obtained by Khatri [8] and Marsaglia [9], gives upper and lower bounds for the rank of the sum of two matrices.

Given a matrix expression with some variant matrices in it, the rank of the matrix expression will vary with respect to the variant matrices. Since the rank of matrix is an integer between 0 and the minimum of row and column numbers of the matrix [10], then the inequalities for ranks of matrix expressions must exist with respect to their variant matrices. Many problems in matrix theory and applications are closely related to the inequalities for ranks of matrix expressions with variant matrices. For example, a matrix expression D A X B of order n is nonsingular if and only if the maximal rank of D A X B with respect to X is n; a matrix equation A X B = C is consistent if and only if the minimal rank of the matrix expression C A X B with respect to X is zero; two consistent matrix equations X 1 = X 1 A X 1 and X 2 = X 2 B X 2 have a common solution if and only if the minimal rank of the difference X 1 X 2 of their solutions is zero. In general, for any two matrix expressions P ( X 1 , X 2 , , X m ) and Q ( Y 1 , Y 2 , , Y n ) of the same size, there are X 1 , X 2 , , X m and Y 1 , Y 2 , , Y n such that P ( X 1 , X 2 , , X m ) = Q ( Y 1 , Y 2 , , Y n ) if and only if
min X 1 , X 2 , , X m ; Y 1 , Y 2 , , Y n r ( P ( X 1 , X 2 , , X m ) Q ( Y 1 , Y 2 , , Y n ) ) = 0 .

The inequalities for ranks of a matrix expression play the important roles in matrix theory for describing the dimension of the row and column vector space of the matrix expressions, which are well understand and are easy to compute by the well-known elementary or congruent matrix expressions, see, e.g., [8, 1115]. The inequalities for ranks of matrix expressions could be regarded as one of the fundamental topics in matrix theory and applications, which can be used to investigate nonsingularity and inverse of a matrix, range and rank invariance of a matrix, relations between subspaces, equalities of matrix expressions with variable matrices, reverse order laws for generalized inverses, existence of solutions to various matrix equations, and so on, see, e.g., [8, 9, 1619].

In this paper, by using the maximal and minimal ranks of generalized Schur complement [11, 14], we get several inequalities for ranks of the matrix expressions D A B X A B , where X is taken, respectively, as B ( 1 ) A ( 1 ) , B ( 1 , 2 ) A ( 1 , 2 ) , B ( 1 , 3 ) A ( 1 , 3 ) , B ( 1 , 4 ) A ( 1 , 4 ) , B ( 1 , 2 , 3 ) A ( 1 , 2 , 3 ) , B ( 1 , 2 , 4 ) A ( 1 , 2 , 4 ) and B A . We also derive various valuable consequences.

In order to find the inequalities for ranks of matrix expressions, we first mention the following lemmas, which will be used in this paper.

Lemma 1.1 [11, 14]

Let A C m × n , B C m × l , C C k × n and D C k × l . Then
max A ( 1 ) r ( D C A ( 1 ) B ) = min { r ( C , D ) , r ( B D ) , r ( A B C D ) r ( A ) } ,
(2)
min A ( 1 ) r ( D C A ( 1 ) B ) = r ( A ) + r ( C , D ) + r ( B D ) + r ( A B C D ) r ( A O O B C D ) r ( A O B O C D ) ,
(3)
max A ( 1 , 2 ) r ( D C A ( 1 , 2 ) B ) = min { r ( A ) + r ( D ) , r ( C , D ) , r ( B D ) , r ( A B C D ) r ( A ) } ,
(4)
min A ( 1 , 2 ) r ( D C A ( 1 , 2 ) B ) = r ( B D ) + r ( C , D ) + r ( A ) + max { S 1 , S 2 } ,
(5)
where
S 1 = r ( A B C D ) r ( A O B O C D ) r ( A O O B C D ) , S 2 = r ( D ) r ( A O C D ) r ( A B O D ) , max A ( 1 , 3 ) r ( D C A ( 1 , 3 ) B ) = min { r ( A A A B C D ) r ( A ) , r ( B D ) } ,
(6)
min A ( 1 , 3 ) r ( D C A ( 1 , 3 ) B ) = r ( A A A B C D ) + r ( B D ) r ( A O O B C D ) ,
(7)
r ( D C A B ) = r ( A A A A B C A D ) r ( A ) .
(8)

Lemma 1.2 [20]

Suppose B, C and D satisfy R ( D ) R ( C ) and R ( D ) R ( B ) . Then the Moore-Penrose inverse of the block matrix
M = ( O B C D )
can be expressed as
M = ( O B C D ) = ( C D B C B O ) .
(9)

Lemma 1.3 [19]

Let A C m × n , B C m × k , C C p × n and D C p × k . Then
  1. (I)

    r ( A , B ) = r ( A ) + r ( E A B ) = r ( E B A ) + r ( B ) ,

     
  2. (II)

    r ( A , B ) = r ( A ) + r ( B ) dim ( R ( A ) R ( B ) ) ,

     
  3. (III)

    r ( A , B ) r ( A ) + r ( B ) ,

     
  4. (IV)

    r ( A C ) = r ( A ) + r ( C F A ) = r ( A F C ) + r ( C ) r ( A ) + r ( C ) ,

     
  5. (V)

    r ( A C ) = r ( A ) + r ( C ) dim ( N ( A ) N ( C ) ) ,

     
  6. (VI)

    r ( A B C D ) = r ( A ) + r ( C F A ) + r ( E A B ) + r ( E C 1 S A F B 1 ) ,

     

where C 1 = C F A , B 1 = E A B , S A = D C A B and dim denotes dimension.

2 Inequalities for ranks of D A B B ( 1 ) A ( 1 ) A B

In this section, we will present several inequalities for ranks of the matrix expression D A B B ( 1 ) A ( 1 ) A B , with respect to two variant matrices B ( 1 ) B { 1 } and A ( 1 ) A { 1 } , where A C m × n , B C n × p and D C m × p are given matrices.

Theorem 2.1 Let A C m × n , B C n × p and D C m × p . Then for any B ( 1 ) B { 1 } and A ( 1 ) A { 1 } , the following inequalities hold:
r ( D A B B ( 1 ) A ( 1 ) A B ) min { r ( A B , D ) , r ( A B D ) , n + r ( A B ) + r ( D A B ) r ( A ) r ( B ) } , r ( D A B B ( 1 ) A ( 1 ) A B ) r ( A B D ) + r ( D A B ) r ( A B ) min { r ( A B D O A B ) r ( A B ) , n + r ( A B D ) r ( A ) r ( B ) } .
Proof Using formula (2) in Lemma 1.1 and formula (IV) in Lemma 1.3, we have
max A ( 1 ) r ( D A B B ( 1 ) A ( 1 ) A B ) = min { r ( A B B ( 1 ) , D ) , r ( A B D ) , r ( A A B A B B ( 1 ) D ) r ( A ) } = min { r ( A B B ( 1 ) , D ) , r ( A B D ) , r ( A B B ( 1 ) F A , D A B ) } = min { r ( A B D ) , r ( A B B ( 1 ) F A , D A B ) } .
(10)
The last equation holds, since
r ( A B B ( 1 ) F A , D A B ) = r ( ( A B B ( 1 ) , D A B ) ( F A O O I p ) ) r ( A B B ( 1 ) , D A B ) = r [ ( A B B ( 1 ) , D A B ) ( I n B O I p ) ] = r ( A B B ( 1 ) , D ) ,

i.e. r ( A B B ( 1 ) F A , D A B ) r ( A B B ( 1 ) , D ) .

Using formula (2) in Lemma 1.1 again, we have
max B ( 1 ) r ( A B B ( 1 ) F A , D A B ) = max B ( 1 ) r ( [ D A B , O ] + A B B ( 1 ) [ O , F A ] ) = min { r ( A B , D A B ) , r ( F A ) + r ( D A B ) , r ( B O F A A B D A B O ) r ( B ) } = min { r ( A B , D ) , r ( F A ) + r ( D A B ) , r ( B , F A ) + r ( D A B ) r ( B ) } = min { r ( A B , D ) , r ( B , F A ) + r ( D A B ) r ( B ) } = min { r ( A B , D ) , n + r ( A B ) + r ( D A B ) r ( A ) r ( B ) } .
(11)
The third equation holds, since from formula (III) in Lemma 1.3,
r ( B , F A ) r ( B ) + r ( F A ) .
Combining (10) with (11), we have
max B ( 1 ) , A ( 1 ) r ( D A B B ( 1 ) A ( 1 ) A B ) = min { r ( A B , D ) , r ( A B D ) , n + r ( A B ) + r ( D A B ) r ( A ) r ( B ) } .
(12)
That is, for any B ( 1 ) B { 1 } and A ( 1 ) A { 1 } , the following inequalities hold:
r ( D A B B ( 1 ) A ( 1 ) A B ) min { r ( A B , D ) , r ( A B D ) , n + r ( A B ) + r ( D A B ) r ( A ) r ( B ) } .
On the other hand, applying formula (3) in Lemma 1.1, we have
min A ( 1 ) r ( D A B B ( 1 ) A ( 1 ) A B ) = r ( A ) + r ( A B B ( 1 ) , D ) + r ( A B D ) + r ( A A B A B B ( 1 ) D ) r ( A O A B O A B B ( 1 ) D ) r ( A O O A B A B B ( 1 ) D ) = r ( A B D ) + r ( A A B A B B ( 1 ) D ) r ( A O O A B A B B ( 1 ) D ) .
(13)
According to (13), we have
min B ( 1 ) , A ( 1 ) r ( D A B B ( 1 ) A ( 1 ) A B ) r ( A B D ) + min B ( 1 ) r ( A A B A B B ( 1 ) D ) max B ( 1 ) r ( A O O A B A B B ( 1 ) D ) .
(14)
By formula (3) in Lemma 1.1 and formula (IV) in Lemma 1.3, we have
min B ( 1 ) r ( A A B A B B ( 1 ) D ) = r ( A ) + min B ( 1 ) r ( A B B ( 1 ) F A , D A B ) = r ( A ) + min B ( 1 ) r ( [ D A B , O ] + A B B ( 1 ) [ O , F A ] ) = r ( A ) + r ( D A B ) .
(15)
Using formula (2) in Lemma 1.1 and formulas (III), (IV) in Lemma 1.3, we have
max B ( 1 ) r ( A O O A B A B B ( 1 ) D ) = r ( A ) + r ( A B ) + max B ( 1 ) r ( A B B ( 1 ) F A , D F A B ) = r ( A ) + r ( A B ) + max B ( 1 ) r ( [ D F A B , O ] + A B B ( 1 ) [ O , F A ] ) = r ( A ) + r ( A B ) + min { r ( A B , D F A B ) , r ( B , F A ) + r ( D F A B ) r ( B ) } = r ( A ) + r ( A B ) + min { r ( A B D O A B ) r ( A B ) , n + r ( A B D ) r ( A ) r ( B ) } .
(16)
Combining the formulas (14), (15) with (16), we obtain
min B ( 1 ) , A ( 1 ) r ( D A B B ( 1 ) A ( 1 ) A B ) r ( A B D ) + r ( D A B ) r ( A B ) min { r ( A B D O A B ) r ( A B ) , n + r ( A B D ) r ( A ) r ( B ) } .
(17)
That is, for any B ( 1 ) B { 1 } and A ( 1 ) A { 1 } , the following inequality holds:
r ( D A B B ( 1 ) A ( 1 ) A B ) r ( A B D ) + r ( D A B ) r ( A B ) min { r ( A B D O A B ) r ( A B ) , n + r ( A B D ) r ( A ) r ( B ) } .

 □

Substituting in Theorem 2.1 with D = A B , we immediately obtain the following corollaries.

Corollary 2.1 ([[21], Theorem 2.2], [[22], Theorem 2.3])

For any matrices A C m × n and B C n × p , the identity A B = A B B ( 1 ) A ( 1 ) A B holds for any B ( 1 ) B { 1 } and A ( 1 ) A { 1 } if and only if
A B = O or n + r ( A B ) = r ( A ) + r ( B ) .

In particular, substituting in Theorem 2.1 with D = O leads to the following result.

Corollary 2.2 Let A C m × n and B C n × p . Then for any B ( 1 ) B { 1 } and A ( 1 ) A { 1 } , the following inequalities holds:
r ( A B ) min { r ( A B ) , r ( A B ) + n r ( A ) r ( B ) } r ( A B B ( 1 ) A ( 1 ) A B ) r ( A B ) .
Corollary 2.3 Let A C m × n and B C n × p . Then the identity
R ( A B B ( 1 ) A ( 1 ) A B ) = R ( A B )
holds, for any B ( 1 ) B { 1 } and A ( 1 ) A { 1 } , if and only if
A B = O or n + r ( A B ) = r ( A ) + r ( B ) .

3 Inequalities for ranks of D A B B ( 1 , 2 ) A ( 1 , 2 ) A B

By analogy with the proof of Theorem 2.1, in this section we will present several inequalities for ranks of the matrix expression D A B B ( 1 , 2 ) A ( 1 , 2 ) A B . The main result in this section is the following theorem.

Theorem 3.1 Let A C m × n , B C n × p and D C m × p . Then for any B ( 1 , 2 ) B { 1 , 2 } and A ( 1 , 2 ) A { 1 , 2 } , the following inequalities hold:
r ( D A B B ( 1 , 2 ) A ( 1 , 2 ) A B ) min { r ( A B D ) , r ( A B , D ) , n + r ( A B ) + r ( D A B ) r ( A ) r ( B ) } , r ( D A B B ( 1 , 2 ) A ( 1 , 2 ) A B ) r ( A ) + r ( A B D ) + r ( A B , D ) + max { T 1 , T 2 } ,
where
T 1 = r ( D A B ) r ( A B , D ) r ( A B ) r ( A ) min { r ( A B D O A B ) r ( A B ) , n + r ( A B D ) r ( A ) r ( B ) } , T 2 = 2 r ( A ) min { r ( A B , D ) , n + r ( A B ) + r ( D ) r ( A ) r ( B ) } .
Proof Applying formula (4) in Lemma 1.1 and the formulas (III), (IV) in Lemma 1.3, we have
max A ( 1 , 2 ) r ( D A B B ( 1 , 2 ) A ( 1 , 2 ) A B ) = min { r ( A ) + r ( D ) , r ( A B B ( 1 , 2 ) , D ) , r ( A B D ) , r ( A A B A B B ( 1 , 2 ) D ) r ( A ) } = min { r ( A B D ) , r ( A B B ( 1 , 2 ) F A , D A B ) } .
(18)
The second equation holds, since r ( A B D ) r ( A B ) + r ( D ) r ( A ) + r ( D ) , and
r ( A B B ( 1 , 2 ) F A , D A B ) = r ( ( A B B ( 1 , 2 ) , D A B ) ( F A O O I p ) ) r ( A B B ( 1 , 2 ) , D A B ) .
Applying the formulas (4) in Lemma 1.1 again, we have
max B ( 1 , 2 ) r ( A B B ( 1 , 2 ) F A , D A B ) = max B ( 1 , 2 ) r ( [ D A B , O ] + A B B ( 1 , 2 ) [ O , F A ] ) = min { r ( A B , D A B ) , r ( F A ) + r ( D A B ) , r ( B ) + r ( D A B ) , r ( B O F A A B D A B O ) r ( B ) } = min { r ( A B , D ) , r ( B , F A ) + r ( D A B ) r ( B ) } = min { r ( A B , D ) , n + r ( A B ) + r ( D A B ) r ( A ) r ( B ) } .
(19)
The third equation holds, since from formula (III) in Lemma 1.3
r ( B , F A ) r ( B ) + r ( F A )
and
r ( A B , D ) = r ( A B , D A B ) r ( B ) + r ( D A B ) .
In view of (18) and (19) it follows that
max B ( 1 , 2 ) , A ( 1 , 2 ) r ( D A B B ( 1 , 2 ) A ( 1 , 2 ) A B ) = min { r ( A B D ) , r ( A B , D ) , n + r ( A B ) + r ( D A B ) r ( A ) r ( B ) } .
(20)
That is, for any B ( 1 , 2 ) B { 1 , 2 } and A ( 1 , 2 ) A { 1 , 2 } , the following inequalities hold:
r ( D A B B ( 1 , 2 ) A ( 1 , 2 ) A B ) min { r ( A B , D ) , r ( A B D ) , n + r ( A B ) + r ( D A B ) r ( A ) r ( B ) } .
On the other hand, using formula (5) in Lemma 1.1 and formula (I) in Lemma 1.3, we have
min A ( 1 , 2 ) r ( D A B B ( 1 , 2 ) A ( 1 , 2 ) A B ) = r ( A B D ) + r ( A B B ( 1 , 2 ) , D ) + r ( A ) + max { S 1 ¯ , S 2 ¯ } = r ( A B D ) + r ( A B , D ) + r ( A ) + max { S 1 ¯ , S 2 ¯ } ,
(21)
where
S 1 ¯ = r ( A A B A B B ( 1 , 2 ) D ) r ( A O A B O A B B ( 1 , 2 ) D ) r ( A O O A B A B B ( 1 , 2 ) D ) , S 2 ¯ = r ( D ) r ( A O A B B ( 1 , 2 ) D ) r ( A A B O D ) ,
and
r ( A B , D ) = r ( ( A B B ( 1 , 2 ) , D ) ( B O O I p ) ) r ( A B B ( 1 , 2 ) , D ) = r ( ( A B , D ) ( B ( 1 , 2 ) O O I p ) ) r ( A B , D ) .
According to the results in (21), we have
min B ( 1 , 2 ) , A ( 1 , 2 ) r ( D A B B ( 1 , 2 ) A ( 1 , 2 ) A B ) r ( A B D ) + r ( A B , D ) + r ( A ) + max { min S 1 ¯ , min S 2 ¯ } .
(22)
Note that, for any B ( 1 , 2 ) B { 1 , 2 } ,
min S 1 ¯ ( min B ( 1 , 2 ) r ( A A B A B B ( 1 , 2 ) D ) ) r ( A ) r ( A B , D ) max B ( 1 , 2 ) r ( A O O A B A B B ( 1 , 2 ) D )
(23)
and
min S 2 ¯ r ( A ) max B ( 1 , 2 ) r ( A O A B B ( 1 , 2 ) D ) .
(24)
Applying formula (5) in Lemma 1.1 and formula (IV) in Lemma 1.3, we have
min B ( 1 , 2 ) r ( A A B A B B ( 1 , 2 ) D ) = r ( A ) + min B ( 1 , 2 ) r ( A B B ( 1 , 2 ) F A , D A B ) = r ( A ) + min B ( 1 , 2 ) r ( [ D A B , O ] + A B B ( 1 , 2 ) [ O , F A ] ) = r ( A ) + r ( D A B )
(25)
and
max B ( 1 , 2 ) r ( A O O A B A B B ( 1 , 2 ) D ) = r ( A ) + r ( A B ) + max B ( 1 , 2 ) r ( A B B ( 1 , 2 ) F A , D F A B ) = r ( A ) + r ( A B ) + max B ( 1 , 2 ) r ( [ D F A B , O ] + A B B ( 1 , 2 ) [ O , F A ] ) = min { r ( A B D O A B ) r ( A B ) , n + r ( A B D ) r ( A ) r ( B ) } + r ( A ) + r ( A B ) .
(26)
Combining (23), (25) with (26), we have
min S 1 ¯ r ( D A B ) r ( A B , D ) r ( A B ) r ( A ) min { r ( A B D O A B ) r ( A B ) , n + r ( A B D ) r ( A ) r ( B ) } .
That is,
min S 1 ¯ T 1 .
(27)
According to formula (4) in Lemma 1.1 and the formulas (III), (IV) in Lemma 1.3, we have
max B ( 1 , 2 ) r ( A O A B B ( 1 , 2 ) D ) = r ( A ) + max B ( 1 , 2 ) r ( A B B ( 1 , 2 ) F A , D ) = r ( A ) + max B ( 1 , 2 ) r ( [ D , O ] + A B B ( 1 , 2 ) [ O , F A ] ) = r ( A ) + min { r ( A B , D ) , r ( B , F A ) + r ( D ) r ( B ) } = r ( A ) + min { r ( A B , D ) , n + r ( A B ) + r ( D ) r ( A ) r ( B ) } .
(28)
By (24) and (28), we have
min S 2 ¯ 2 r ( A ) min { r ( A B , D ) , n + r ( A B ) + r ( D ) r ( A ) r ( B ) } .
That is,
min S 2 ¯ T 2 .
(29)
Finally on account of (22), (27), and (29), it is seen that
min B ( 1 , 2 ) , A ( 1 , 2 ) r ( D A B B ( 1 , 2 ) A ( 1 , 2 ) A B ) r ( A B D ) + r ( A B , D ) + r ( A ) + max { T 1 , T 2 } .
(30)
That is, for any B ( 1 , 2 ) B { 1 , 2 } and A ( 1 , 2 ) A { 1 , 2 } , we have
r ( D A B B ( 1 , 2 ) A ( 1 , 2 ) A B ) r ( A ) + r ( A B D ) + r ( A B , D ) + max { T 1 , T 2 } .

 □

From Theorem 3.1, we immediately obtain the following corollaries by the formulas (20) and (30).

Corollary 3.1 Let A C m × n and B C n × p . Then the identity
A B = A B B ( 1 , 2 ) A ( 1 , 2 ) A B ,
holds for any B ( 1 , 2 ) B { 1 , 2 } and A ( 1 , 2 ) A { 1 , 2 } if and only if
A B = O or n + r ( A B ) = r ( A ) + r ( B ) .

In particular, substituting in Theorem 3.1 with D = O leads to the following.

Corollary 3.2 Let A C m × n and B C n × p . Then for any B ( 1 , 2 ) B { 1 , 2 } and A ( 1 , 2 ) A { 1 , 2 } , the following inequalities holds:
r ( A B ) min { r ( A B ) , r ( A B ) + n r ( A ) r ( B ) } r ( A B B ( 1 , 2 ) A ( 1 , 2 ) A B ) r ( A B ) .
Corollary 3.3 Let A C m × n and B C n × p . Then the identity
R ( A B B ( 1 , 2 ) A ( 1 , 2 ) A B ) = R ( A B )
holds for any B ( 1 , 2 ) B { 1 , 2 } and A ( 1 , 2 ) A { 1 , 2 } if and only if
A B = O or n + r ( A B ) = r ( A ) + r ( B ) .

4 Inequalities for ranks of D A B B ( 1 , 3 ) A ( 1 , 3 ) A B and D A B B ( 1 , 4 ) A ( 1 , 4 ) A B

Applying the formulas (6) and (7) in Lemma 1.1 to the matrix expressions D A B B ( 1 , 3 ) A ( 1 , 3 ) A B and D A B B ( 1 , 4 ) A ( 1 , 4 ) A B , we obtain some inequalities for ranks of this two matrix expressions. The main result in this section is the following theorem.

Theorem 4.1 Let A C m × n , B C n × p and D C m × p . Then for any B ( 1 , 3 ) B { 1 , 3 } and A ( 1 , 3 ) A { 1 , 3 } , the following inequalities hold:
r ( D A B B ( 1 , 3 ) A ( 1 , 3 ) A B ) min { r ( B B O B A B D A B O O O A ) r ( A ) r ( B ) , r ( A B D ) } , r ( D A B B ( 1 , 3 ) A ( 1 , 3 ) A B ) r ( B B O B A B D A B O O O A ) + r ( A B D ) r ( A B O D B B B O O A O O O A B ) .
Proof According to formula (6) in Lemma 1.1 and formula (IV) in Lemma 1.3, we have
max A ( 1 , 3 ) r ( D A B B ( 1 , 3 ) A ( 1 , 3 ) A B ) = min { r ( A B D ) , r ( A A A A B A B B ( 1 , 3 ) D ) r ( A ) } = min { r ( A B D ) , r ( A B B ( 1 , 3 ) F A , D A B ) } .
(31)
Applying formula (6) in Lemma 1.1 and formula (IV) in Lemma 1.3 again, we have
max B ( 1 , 3 ) r ( A B B ( 1 , 3 ) F A , D A B ) = max B ( 1 , 3 ) r ( [ D A B , O ] + A B B ( 1 , 3 ) [ O , F A ] ) = min { r ( F A ) + r ( D A B ) , r ( B B B F A O A B O D A B ) r ( B ) } = r ( B B O B A B D A B O O O A ) r ( A ) r ( B ) .
(32)
The third equation holds, since from formula (III) in Lemma 1.3,
r ( B B B F A O A B O D A B ) r ( B B B F A A B O ) + r ( D A B ) r ( B ) + r ( B F A ) + r ( D A B ) r ( B ) + r ( F A ) + r ( D A B ) .
Combining (31) with (32), we have
max B ( 1 , 3 ) , A ( 1 , 3 ) r ( D A B B ( 1 , 3 ) A ( 1 , 3 ) A B ) = min { r ( B B O B A B D A B O O O A ) r ( A ) r ( B ) , r ( A B D ) } .
(33)
That is, for any B ( 1 , 3 ) B { 1 , 3 } and A ( 1 , 3 ) A { 1 , 3 } , the following inequalities hold:
r ( D A B B ( 1 , 3 ) A ( 1 , 3 ) A B ) min { r ( B B O B A B D A B O O O A ) r ( A ) r ( B ) , r ( A B D ) } .
On the other hand, from formula (7) in Lemma 1.1 and formula (IV) in Lemma 1.3, we have
min A ( 1 , 3 ) r ( D A B B ( 1 , 3 ) A ( 1 , 3 ) A B ) = r ( A B D ) + r ( A A A A B A B B ( 1 , 3 ) D ) r ( A O O A B A B B ( 1 , 3 ) D ) = r ( A B B ( 1 , 3 ) F A , D A B ) + r ( A B D ) r ( A B ) r ( A B B ( 1 , 3 ) F A , D F A B ) .
(34)
From (34), we get
min B ( 1 , 3 ) , A ( 1 , 3 ) r ( D A B B ( 1 , 3 ) A ( 1 , 3 ) A B ) min B ( 1 , 3 ) { r ( A B B ( 1 , 3 ) F A , D A B ) } + r ( A B D ) r ( A B ) max B ( 1 , 3 ) r ( A B B ( 1 , 3 ) F A , D F A B ) .
(35)
Applying formula (7) in Lemma 1.1 and formula (IV) in Lemma 1.3 again, we have
min B ( 1 , 3 ) r ( A B B ( 1 , 3 ) F A , D A B ) = min B ( 1 , 3 ) r ( [ D A B , O ] + A B B ( 1 , 3 ) [ O , F A ] ) = r ( D A B ) r ( B O A B D A B ) + r ( B B B F A O A B O D A B ) = r ( B B O B A B D A B O O O A ) r ( A ) r ( B ) .
(36)
Applying formula (6) in Lemma 1.1 and the formulas (I), (III) in Lemma 1.3, we have
max B ( 1 , 3 ) r ( A B B ( 1 , 3 ) F A , D F A B ) = max B ( 1 , 3 ) r ( [ D F A B , O ] + A B B ( 1 , 3 ) [ O , F A ] ) = min { r ( B B B F A O A B O D F A B ) r ( B ) , r ( F A ) + r ( D F A B ) } = r ( B B B F A O A B O D F A B ) r ( B ) = r ( B B B O A B O D O A O O O A B ) r ( A B ) r ( A ) r ( B ) .
(37)
The third equation holds, since
r ( B B B F A O A B O D F A B ) r ( B B B F A A B O ) + r ( D F A B ) r ( B ) + r ( B F A ) + r ( D F A B ) r ( B ) + r ( F A ) + r ( D F A B ) .
By (35), (36), and (37), we have
min B ( 1 , 3 ) , A ( 1 , 3 ) r ( D A B B ( 1 , 3 ) A ( 1 , 3 ) A B ) r ( B B O B A B D A B O O O A ) + r ( A B D ) r ( A B O D B B B O O A O O O A B ) .
(38)
That is, for any B ( 1 , 3 ) B { 1 , 3 } and A ( 1 , 3 ) A { 1 , 3 } , we have
r ( D A B B ( 1 , 3 ) A ( 1 , 3 ) A B ) r ( B B O B A B D A B O O O A ) + r ( A B D ) r ( A B O D B B B O O A O O O A B ) .

 □

From Theorem 4.1, we immediately obtain the following corollaries by (33) and (38).

Corollary 4.1 Let A C m × n and B C n × p . Then the identity
A B = A B B ( 1 , 3 ) A ( 1 , 3 ) A B
holds for any B ( 1 , 3 ) B { 1 , 3 } and A ( 1 , 3 ) A { 1 , 3 } if and only if
r ( A B ) + r ( B A ) = r ( A ) + r ( B ) .
Corollary 4.2 Let A C m × n and B C n × p . Then for any B ( 1 , 3 ) B { 1 , 3 } and A ( 1 , 3 ) A { 1 , 3 } , the following inequalities hold:
r ( A ) + r ( B ) r ( B A ) r ( A B B ( 1 , 3 ) A ( 1 , 3 ) A B ) r ( A B ) .
Corollary 4.3 Let A C m × n and B C n × p . Then the identity
R ( A B B ( 1 , 3 ) A ( 1 , 3 ) A B ) = R ( A B )
holds for any B ( 1 , 3 ) B { 1 , 3 } and A ( 1 , 3 ) A { 1 , 3 } if and only if
r ( A ) + r ( B ) r ( B A ) = r ( A B ) .

Notice that a matrix X belongs to A { 1 , 4 } if and only if X belongs to A { 1 , 3 } . So from the results obtained in above of Section 4, we can get the inequalities for ranks of D A B B ( 1 , 4 ) A ( 1 , 4 ) A B . We state the following theorems without proofs.

Theorem 4.2 Let A C m × n , B C n × p and D C m × p . Then for any B ( 1 , 4 ) B { 1 , 4 } and A ( 1 , 4 ) A { 1 , 4 } , the following inequalities hold:
r ( D A B B ( 1 , 4 ) A ( 1 , 4 ) A B ) min { r ( A A O A B A B O O O D A B ) r ( A ) r ( B ) , r ( A B , D ) } , r ( D A B B ( 1 , 4 ) A ( 1 , 4 ) A B ) r ( A A O A B A B O O O D A B ) + r ( A B , D ) r ( A A O O A B A B O O O O A B D ) .
Corollary 4.4 Let A C m × n and B C n × p . Then the identity
A B = A B B ( 1 , 4 ) A ( 1 , 4 ) A B
holds for any B ( 1 , 4 ) B { 1 , 4 } and A ( 1 , 4 ) A { 1 , 4 } if and only if
r ( A B ) + r ( A , B ) = r ( A ) + r ( B ) .
Corollary 4.5 Let A C m × n and B C n × p . Then for any B ( 1 , 4 ) B { 1 , 4 } and A ( 1 , 4 ) A { 1 , 4 } , the following inequalities hold:
r ( A ) + r ( B ) r ( A , B ) r ( A B B ( 1 , 3 ) A ( 1 , 3 ) A B ) r ( A B ) .
Corollary 4.6 Let A C m × n and B C n × p . Then the identity
R ( A B B ( 1 , 4 ) A ( 1 , 4 ) A B ) = R ( A B )
holds for any B ( 1 , 4 ) B { 1 , 4 } and A ( 1 , 4 ) A { 1 , 4 } if and only if
r ( A ) + r ( B ) r ( A , B ) = r ( A B ) .

5 Inequalities for ranks of D A B B ( 1 , 2 , 3 ) A ( 1 , 2 , 3 ) A B and D A B B ( 1 , 2 , 4 ) A ( 1 , 2 , 4 ) A B

From the results in [3], it is seen that a matrix X belongs to B { 1 , 2 , 3 } if and only if
X = ( B B ) ( 1 ) B ,
where ( B B ) ( 1 ) ( B B ) { 1 } is arbitrary. Similarly, a matrix X belongs to B { 1 , 2 , 4 } if and only if
X = B ( B B ) ( 1 ) ,
where ( B B ) ( 1 ) ( B B ) { 1 } is arbitrary. Thus
D A B B ( 1 , 2 , 3 ) A ( 1 , 2 , 3 ) A B = D A B ( B B ) ( 1 ) B ( A A ) ( 1 ) A A B
(39)
and
D A B B ( 1 , 2 , 4 ) A ( 1 , 2 , 4 ) A B = D A B B ( B B ) ( 1 ) A ( A A ) ( 1 ) A B .
(40)

Applying the formulas (2) and (3) in Lemma 1.1 to (39) and (40), we have the following theorems, which can be shown by a similar approach to Theorem 2.1, and the proof are omitted here.

Theorem 5.1 Let A C m × n , B C n × p and D C m × p . Then for any B ( 1 , 2 , 3 ) B { 1 , 2 , 3 } and A ( 1 , 2 , 3 ) A { 1 , 2 , 3 } , the following inequalities hold:
r ( D A B B ( 1 , 2 , 3 ) A ( 1 , 2 , 3 ) A B ) min { r ( B B O B A B D A B O O O A ) r ( A ) r ( B ) , r ( A B D ) } , r ( D A B B ( 1 , 2 , 3 ) A ( 1 , 2 , 3 ) A B ) r ( B B O B A B D A B O O O A ) + r ( A B D ) r ( A B O D B B B O O A O O O A B ) .
Theorem 5.2 Let A C m × n , B C n × p and D C m × p . Then for any B ( 1 , 2 , 4 ) B { 1 , 2 , 4 } and A ( 1 , 2 , 4 ) A { 1 , 2 , 4 } , the following inequalities hold:
r ( D A B B ( 1 , 2 , 4 ) A ( 1 , 2 , 4 ) A B ) min { r ( A A O A B A B O O O D A B ) r ( A ) r ( B ) , r ( A B , D ) } , r ( D A B B ( 1 , 2 , 4 ) A ( 1 , 2 , 4 ) A B ) r ( A A O A B A B O O O D A B ) + r ( A B , D ) r ( A A O O A B A B O O O O A B D ) .

6 Rank of D A B B A A B

In this section, we will present the rank of the linear matrix expression
D A B B A A B ,

where A C m × n , B C n × p and D C m × p are given matrices.

Theorem 6.1 Let A C m × n , B C n × p and D C m × p . Then
r ( D A B B A A B ) = r ( O A A A B B B B A O A B O D ) r ( A ) r ( B ) .
Proof Let
T = ( O A A B B B A ) .
(41)
Then applying formula (9) in Lemma 1.2, we have
T = ( B A ( B B ) ( A A ) O ) , T T = ( A A O O B B ) , T T = ( B B O O A A ) .
(42)
The sub-matrix in the upper left corner of the Moore-Penrose inverse of T can be expressed as
B A = E 1 T E 2 ,
where E 1 = ( I p , O ) and E 2 = ( I m , O ) . Hence
D A B B A A B = D + A B E 1 T E 2 A B .
(43)
Applying formula (8) in Lemma 1.1, we have
r ( D A B B A A B ) = r ( D + A B E 1 T E 2 A B ) = r ( T T T T E 2 A B A B E 1 T D ) r ( T ) = r ( T E 2 A B A B E 1 D ) r ( T ) = r ( O A A A B B B B A O A B O D ) r ( A ) r ( B ) .

 □

As a direct consequence of Theorem 6.1, we immediately get the following results.

Corollary 6.1 Let A C m × n and B C n × p . Then the identity A B = A B B A A B holds if and only if
r ( A B A A B B B A ) + r ( A B ) = r ( A ) + r ( B ) .
Corollary 6.2 Let A C m × n and B C n × p . Then the identity R ( A B B A A B ) = R ( A B ) holds if and only if
r ( O A A A B B B B A O A B O O ) = r ( A B ) + r ( A ) + r ( B ) .

Declarations

Acknowledgements

The author would like to thank the Editor-in-Chief and the anonymous referees for their very detailed comments, which greatly improved the presentation of this article. The work was supported by the NSFC (Grant No: 11301397) and the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (2012LYM-0126) and the Basic Theory and Scientific Research of Science and Technology Project of Jiangmen City, China, 2013.

Authors’ Affiliations

(1)
School of Mathematics and Computational Science, Wuyi University

References

  1. Penrose R: A generalized inverse for matrices. Proc. Camb. Philos. Soc. 1955, 51: 406-413. 10.1017/S0305004100030401MATHMathSciNetView ArticleGoogle Scholar
  2. Ben-Israel A, Greville TNE: Generalized Inverse: Theory and Applications. Wiley-Interscience, New York; 1974. 2nd edn. Springer, New York (2002)MATHGoogle Scholar
  3. Rao CR, Mitra SK: Generalized Inverse of Matrices and Its Applications. Wiley, New York; 1971.MATHGoogle Scholar
  4. Wang G, Wei Y, Qiao S: Generalized Inverses: Theory and Computations. Science Press, Beijing; 2004.Google Scholar
  5. Mirsky L: An Introduction to Linear Algebra. Oxford University Press, Oxford; 1955.MATHGoogle Scholar
  6. Frobenius G III. In Über den rang einer matrix. Springer, Berlin; 1968:479-490.Google Scholar
  7. Sylvester JJ IV. In The Collected Mathematical Papers of James Joseph Sylvester. Cambridge University Press, Cambridge; 1912:133-145.Google Scholar
  8. Khatri CG: A simplified approach to the derivation of the theorems on the rank of a matrix. J. Maharaja Sayajirao Univ. Baroda 1961, 10: 1-5.Google Scholar
  9. Marsaglia G: Bounds on the rank of the sum of matrices. In Trans. Fourth Prague Conf. on Information Theory, Statistical Decision Functions, Random Processes. Academia, Prague; 1967:455-462.Google Scholar
  10. David CL: Linear Algebra and Its Applications. Addison-Wesley, Reading; 1994.Google Scholar
  11. Tian Y: Upper and lower bounds for ranks of matrix expressions using generalized inverses. Linear Algebra Appl. 2002, 355: 187-214. 10.1016/S0024-3795(02)00345-2MATHMathSciNetView ArticleGoogle Scholar
  12. Tian Y:Ranks of solutions of the matrix equation A X B = C . Linear Multilinear Algebra 2003, 51: 111-125. 10.1080/0308108031000114631MathSciNetView ArticleGoogle Scholar
  13. Tian Y, Cheng S:The maximal and minimal ranks of A B X C with applications. N.Y. J. Math. 2003, 9: 345-362.MATHMathSciNetGoogle Scholar
  14. Tian Y: More on maximal and minimal ranks of Schur complements with applications. Appl. Math. Comput. 2004, 152: 675-692. 10.1016/S0096-3003(03)00585-XMATHMathSciNetView ArticleGoogle Scholar
  15. Tian Y: The maximal and minimal ranks of a quadratic matrix expression with applications. Linear Multilinear Algebra 2011, 59: 627-644. 10.1080/03081081003774268MATHMathSciNetView ArticleGoogle Scholar
  16. Groß J: Comment on range invariance of matrix products. Linear Multilinear Algebra 1996, 41: 157-160. 10.1080/03081089608818469MATHView ArticleGoogle Scholar
  17. Groß J, Tian Y: Invariance properties of a triple matrix product involving generalized inverses. Linear Algebra Appl. 2006, 417: 94-107. 10.1016/j.laa.2006.03.026MATHMathSciNetView ArticleGoogle Scholar
  18. Johnson CR, Whitney GT: Minimum rank completions. Linear Multilinear Algebra 1991, 28: 271-273. 10.1080/03081089108818051MATHMathSciNetView ArticleGoogle Scholar
  19. Marsaglia G, Styan GPH: Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 1974, 2: 269-292. 10.1080/03081087408817070MathSciNetView ArticleGoogle Scholar
  20. Hartwig RE: Block generalized inverses. Arch. Ration. Mech. Anal. 1976, 61: 197-251. 10.1007/BF00281485MATHMathSciNetView ArticleGoogle Scholar
  21. Wei M: Equivalent conditions for generalized inverses of products. Linear Algebra Appl. 1997, 266: 347-363.MATHMathSciNetView ArticleGoogle Scholar
  22. Werner HJ: When is B A a generalized inverse of AB ? Linear Algebra Appl. 1994, 210: 255-263.MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Xiong; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.