Some Results on Characterizations of Matrix Partial Orderings

Abstract

Some characterizations of the left-star, right-star, and star partial orderings between matrices of the same size are obtained. Based on those results, several characterizations of the star partial ordering between EP matrices are given. At last, one characterization of the sharp partial ordering between group matrices is obtained.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Academic Editor:Yang Zhang

Department of Mathematics, Huainan Normal University, Anhui 232001, China

Received 19 February 2014; Accepted 21 April 2014; 28 May 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In this paper we use the following notation. Let ^Cm×n be the set of complex m×n matrices. For any matrix A∈^Cm×n , ^A* , R(A) , and r(A) denote the conjugate transpose, the range, and the rank of A , respectively. The symbol _In denotes the n × n identity matrix, and 0 denotes a zero matrix of appropriate size. The Moore-Penrose inverse of a matrix A∈^Cm×n , denoted by ^A[dagger] , is defined to be the unique matrix X∈^Cn×m satisfying the four matrix equations [figure omitted; refer to PDF] and ^A- denotes any solution to the matrix equation AXA=A with respect to X ; A{1} denotes the set of ^A- ; that is, A{1}={X|"AXA=A} . Moreover, ^A# denotes the group inverse of A with r(^A2 )=r(A) , that is, the unique solution to [figure omitted; refer to PDF] It is well known that ^A# exists if and only if r(^A2 )=r(A) , where case A is also called a group matrix. A matrix A is EP if and only if A is a group matrix with ^A# =^A[dagger] . The symbols ^CGPn and ^CEP...n stand for the subset of ^Cn×n consisting of group matrices and EP matrices, respectively (see, e.g., [1, 2] for details).

Five matrix partial orderings defined in ^Cm×n are considered in this paper. The first of them is the minus partial ordering defined by Hartwig [3] and Nambooripad [4] independently in 1980: [figure omitted; refer to PDF] where ^A- ,^A= ∈A{1} . In [3] it was shown that [figure omitted; refer to PDF] The rank equality indicates why the minus partial ordering is also called the rank-subtractivity partial ordering. In the same paper [3] it was also shown that [figure omitted; refer to PDF] where ^B- ∈B{1} .

The second partial ordering of interest is the star partial ordering introduced by Drazin [5], which is determined by [figure omitted; refer to PDF] It is well known that [figure omitted; refer to PDF]

In 1991, Baksalary and Mitra [6] defined the left-star and right-star partial orderings characterized as [figure omitted; refer to PDF]

The last partial ordering we will deal with in this paper is the sharp partial ordering, introduced by Mitra [7] in 1987, and is defined in the set ^CGPn by [figure omitted; refer to PDF] A detailed discussion of partial orderings and their applications can be found in [1, 8-10].

It is well known that rank of matrix is an important tool in matrix theory and its applications, and many problems are closely related with the ranks of some matrix expressions under some restrictions (see [11-15] for details). Our aim in this paper is to characterize the left-star, right-star, star, and sharp partial orderings by applying rank equalities. In the following, when A is considered below B with respect to one partial ordering, then the partial ordering should entail the assumption r(A)>r(B)...5;1 .

2. The Star Partial Ordering

Let A and B be m×n complex matrices with ranks a and b , respectively. Let A...4;*B . Then there exist unitary matrices U∈^Cm×m and V∈^Cn×n such that [figure omitted; refer to PDF] where both the a×a matrix _Da and the (b-a)×(b-a) matrix D are real, diagonal, and positive definite (see [16, Theorem 2]). In [1, Theorem 5.2.8], it was also shown that [figure omitted; refer to PDF] In [17], Wang obtained the following characterizations of the left-star and right-star partial orderings for matrices: [figure omitted; refer to PDF]

Theorem 1.

Let A,B∈^Cm×n . Then

(i) [figure omitted; refer to PDF]

(ii) [figure omitted; refer to PDF]

(iii): [figure omitted; refer to PDF]

(iv) [figure omitted; refer to PDF]

Proof.

From [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] Applying (12) gives (i).

In the same way, applying [figure omitted; refer to PDF] and (13) gives (ii).

If [figure omitted; refer to PDF] then [figure omitted; refer to PDF] Applying (i), (ii), and (14), we obtain A...4;*B . Conversely, if A...4;*B , by using (11) and (14), we have ^A[dagger] A-^B[dagger] A=0 , and [figure omitted; refer to PDF] Hence, we have (iii).

Similarly, applying A...4;*B , (11), and (14), we obtain A^A[dagger] -A^B[dagger] =0 , A^B[dagger] =^{(AB[dagger] )*} =^{(B* )[dagger]A*} , and [figure omitted; refer to PDF] Then, we obtain (iv).

In [9, Theorem 2.1], Benítez et al. deduce the characterizations of the left-star, right-star, and star partial orderings for matrices, when at least one of the two involved matrices is EP. When both A∈^Cn×n and B∈^Cn×n are EP matrices, [1, Theorems 5.4.15 and 5.4.2] give the following results: [figure omitted; refer to PDF] In addition, it was also shown that A...4;*B if and only if A and B have the form [figure omitted; refer to PDF] where T∈^Cr(A)×r(A) is nonsingular, K∈^{C(r(B)-r(A))×(r(B)-r(A))} is nonsingular, and U∈^Cn×n is unitary (see [1, Theorem 5.4.1]).

Based on these results, we consider the characterizations of the star partial ordering for matrices in the set of ^CEP...n .

Theorem 2.

Let A,B∈^CEP...n , r(B)...5;r(A) . Then

(v) [figure omitted; refer to PDF]

(vi) [figure omitted; refer to PDF]

Proof.

By A,B∈^CEP...n , it is obvious that A^A[dagger] =^A[dagger] A and B^B[dagger] =^B[dagger] B . Then [figure omitted; refer to PDF] Hence, we have (v).

The proof of (vi) is similar to that of (v).

Theorem 3.

Let A,B∈^CEP...n . Then

(vii): [figure omitted; refer to PDF]

(viii): [figure omitted; refer to PDF]

(ix) [figure omitted; refer to PDF]

(x) [figure omitted; refer to PDF]

(xi) [figure omitted; refer to PDF]

Proof.

By A,B∈^CEP...n , it is obvious that A^A[dagger] =^A[dagger] A and B^B[dagger] =^B[dagger] B . Applying (i), (ii), and the rank equality in (vii) we obtain [figure omitted; refer to PDF] that is, A...4;*B . Conversely, suppose that A...4;*B . Applying A-A^A[dagger] B=0 and ^B* B^B[dagger] =^B* , we obtain [figure omitted; refer to PDF]

Applying (11), we obtain ^B* B^B[dagger] B=^B* B and ^B* B^A[dagger] A=^A* A and also ^{(B* B)[dagger]B*} B=^B[dagger] B and ^{(B* B)[dagger]A*} A=^A[dagger] A . Then [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] Hence, we have (viii).

Suppose that A...4;*B . Since A,B∈^CEP...n , applying (27), it is easy to check the rank equality in (ix). Conversely, under the rank equality in (ix), we have [figure omitted; refer to PDF] Since A is EP, there exists a unitary matrix _U1 ∈^Cn×n and a nonsingular matrix T∈^Cr(A)×r(A) such that [figure omitted; refer to PDF] Correspondingly denote ^P-1 BP by [figure omitted; refer to PDF] where _B1 ∈^Cr(A)×r(A) . It follows that [figure omitted; refer to PDF] Since T is a unitary matrix, [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] Since B is EP, _B4 is EP, and there exists a unitary matrix _U2 ∈^{C(n-r(A))×(n-r(A))} and a nonsingular matrix K∈^{C(r(B)-r(A))×(r(B)-r(A))} such that [figure omitted; refer to PDF] Write [figure omitted; refer to PDF] Then A and B have the form [figure omitted; refer to PDF] Applying (27), we have A...4;*B .

The proofs of (x) and (xi) are similar to that of (ix).

3. The Sharp Partial Ordering

Let A,B∈^CGPn with ranks a and b , respectively. It is well known that [figure omitted; refer to PDF] In addition, A ^...4;# B if and only if A and B can be written as [figure omitted; refer to PDF] where E∈^Ca×a is nonsingular, ^{E[variant prime]} ∈^{C(b-a)×(b-a)} is nonsingular, and P∈^Cn×n is nonsingular (see [18]).

In Theorem 4, we give one characterization of the sharp partial ordering by using one rank equality.

Theorem 4.

Let A,B∈^CGPn . Then [figure omitted; refer to PDF]

Proof.

Let A have the core-nilpotent decomposition (see [19, Exercise 5.10.12]) [figure omitted; refer to PDF] with nonsingular matrices Σ∈^Cr(A)×r(A) and P∈^Cn×n . Correspondingly denote ^P-1 BP by [figure omitted; refer to PDF] where _B1 ∈^Cr(A)×r(A) . It follows that [figure omitted; refer to PDF]

Applying (54) to the rank equality in (51), we obtain [figure omitted; refer to PDF] Hence r(Σ_B1 Σ)=r(Σ) , Σ_B2 =0 , _B3 Σ=0 , and Σ_B1 Σ=Σ_B1^Σ-1_B1 Σ . Since Σ∈^Cr(A)×r(A) is invertible and _B1 ∈^Cr(A)×r(A) , it follows immediately that [figure omitted; refer to PDF] Therefore [figure omitted; refer to PDF] Applying [figure omitted; refer to PDF] and (49), we obtain that A ^...4;# B .

Conversely, it is a simple matter.

Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions. The work of the first author was supported in part by the Foundation of Anhui Educational Committee (Grant no. KJ2012B175) and the National Natural Science Foundation of China (Grant no. 11301529). The work of the second author was supported in part by the Foundation of Anhui Educational Committee (Grant no. KJ2013B256).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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