Skip to main content

Inequalities for ranks of matrix expressions involving generalized inverses

Abstract

In this paper, we present several inequalities for ranks of the matrix expressions DABXAB 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.

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 DAXB of order n is nonsingular if and only if the maximal rank of DAXB with respect to X is n; a matrix equation AXB=C is consistent if and only if the minimal rank of the matrix expression CAXB 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 DABXAB, 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 =DC A B and dim denotes dimension.

2 Inequalities for ranks of DAB B ( 1 ) A ( 1 ) AB

In this section, we will present several inequalities for ranks of the matrix expression DAB B ( 1 ) A ( 1 ) AB, 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(AB B ( 1 ) F A ,DAB)r(AB 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=AB, 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 AB=AB B ( 1 ) A ( 1 ) AB holds for any B ( 1 ) B{1} and A ( 1 ) A{1} if and only if

AB=Oorn+r(AB)=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(AB)min { r ( A B ) , r ( A B ) + n r ( A ) r ( B ) } r ( A B B ( 1 ) A ( 1 ) A B ) r(AB).

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(AB)

holds, for any B ( 1 ) B{1} and A ( 1 ) A{1}, if and only if

AB=Oorn+r(AB)=r(A)+r(B).

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

By analogy with the proof of Theorem 2.1, in this section we will present several inequalities for ranks of the matrix expression DAB B ( 1 , 2 ) A ( 1 , 2 ) AB. 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(AB)+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(AB,D)=r(AB,DAB)r(B)+r(DAB).

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(AB,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 ¯ 2r(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(AB,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(AB,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

AB=AB B ( 1 , 2 ) A ( 1 , 2 ) AB,

holds for any B ( 1 , 2 ) B{1,2} and A ( 1 , 2 ) A{1,2} if and only if

AB=Oorn+r(AB)=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(AB)min { r ( A B ) , r ( A B ) + n r ( A ) r ( B ) } r ( A B B ( 1 , 2 ) A ( 1 , 2 ) A B ) r(AB).

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(AB)

holds for any B ( 1 , 2 ) B{1,2} and A ( 1 , 2 ) A{1,2} if and only if

AB=Oorn+r(AB)=r(A)+r(B).

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

Applying the formulas (6) and (7) in Lemma 1.1 to the matrix expressions DAB B ( 1 , 3 ) A ( 1 , 3 ) AB and DAB B ( 1 , 4 ) A ( 1 , 4 ) AB, 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

AB=AB B ( 1 , 3 ) A ( 1 , 3 ) AB

holds for any B ( 1 , 3 ) B{1,3} and A ( 1 , 3 ) A{1,3} if and only if

r(AB)+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(AB).

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(AB)

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(AB).

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 DAB B ( 1 , 4 ) A ( 1 , 4 ) AB. 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

AB=AB B ( 1 , 4 ) A ( 1 , 4 ) AB

holds for any B ( 1 , 4 ) B{1,4} and A ( 1 , 4 ) A{1,4} if and only if

r(AB)+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(AB).

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(AB)

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(AB).

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

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

DAB B ( 1 , 2 , 3 ) A ( 1 , 2 , 3 ) AB=DAB ( B B ) ( 1 ) B ( A A ) ( 1 ) A AB
(39)

and

DAB B ( 1 , 2 , 4 ) A ( 1 , 2 , 4 ) AB=DAB B ( B B ) ( 1 ) A ( A A ) ( 1 ) AB.
(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 DAB B A AB

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

DAB B A AB,

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

DAB B A AB=D+AB E 1 T E 2 AB.
(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 AB=AB B A AB holds if and only if

r( A B A A B B B A )+r(AB)=r(A)+r(B).

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

r( O A A A B B B B A O A B O O )=r(AB)+r(A)+r(B).

References

  1. Penrose R: A generalized inverse for matrices. Proc. Camb. Philos. Soc. 1955, 51: 406-413. 10.1017/S0305004100030401

    MATH  MathSciNet  Article  Google Scholar 

  2. Ben-Israel A, Greville TNE: Generalized Inverse: Theory and Applications. Wiley-Interscience, New York; 1974. 2nd edn. Springer, New York (2002)

    MATH  Google Scholar 

  3. Rao CR, Mitra SK: Generalized Inverse of Matrices and Its Applications. Wiley, New York; 1971.

    MATH  Google 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.

    MATH  Google 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)00345-2

    MATH  MathSciNet  Article  Google Scholar 

  12. Tian Y:Ranks of solutions of the matrix equation AXB=C. Linear Multilinear Algebra 2003, 51: 111-125. 10.1080/0308108031000114631

    MathSciNet  Article  Google Scholar 

  13. Tian Y, Cheng S:The maximal and minimal ranks of ABXC with applications. N.Y. J. Math. 2003, 9: 345-362.

    MATH  MathSciNet  Google 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-X

    MATH  MathSciNet  Article  Google Scholar 

  15. Tian Y: The maximal and minimal ranks of a quadratic matrix expression with applications. Linear Multilinear Algebra 2011, 59: 627-644. 10.1080/03081081003774268

    MATH  MathSciNet  Article  Google Scholar 

  16. Groß J: Comment on range invariance of matrix products. Linear Multilinear Algebra 1996, 41: 157-160. 10.1080/03081089608818469

    MATH  Article  Google 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.2006.03.026

    MATH  MathSciNet  Article  Google Scholar 

  18. Johnson CR, Whitney GT: Minimum rank completions. Linear Multilinear Algebra 1991, 28: 271-273. 10.1080/03081089108818051

    MATH  MathSciNet  Article  Google Scholar 

  19. Marsaglia G, Styan GPH: Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 1974, 2: 269-292. 10.1080/03081087408817070

    MathSciNet  Article  Google Scholar 

  20. Hartwig RE: Block generalized inverses. Arch. Ration. Mech. Anal. 1976, 61: 197-251. 10.1007/BF00281485

    MATH  MathSciNet  Article  Google Scholar 

  21. Wei M: Equivalent conditions for generalized inverses of products. Linear Algebra Appl. 1997, 266: 347-363.

    MATH  MathSciNet  Article  Google Scholar 

  22. Werner HJ: When is B A a generalized inverse of AB ? Linear Algebra Appl. 1994, 210: 255-263.

    MATH  MathSciNet  Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zhiping Xiong.

Additional information

Competing interests

The author declares that they have no competing interests.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Xiong, Z. Inequalities for ranks of matrix expressions involving generalized inverses. J Inequal Appl 2014, 87 (2014). https://doi.org/10.1186/1029-242X-2014-87

Download citation

  • Received:

  • Accepted:

  • Published:

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

Keywords

  • matrix expression
  • generalized inverse
  • inequality
  • rank
  • generalized Schur complement