Abstract and Applied Analysis

Volume 2013 (2013), Article ID 267035, 6 pages

http://dx.doi.org/10.1155/2013/267035

Research Article

Upper and Lower Bounds for Ranks of the Matrix Expression

Zhiping Xiong

School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, China

Received 2 September 2013; Accepted 18 December 2013

Academic Editor: Changbum Chun

Copyright © 2013 Zhiping Xiong. 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.

Abstract.

We consider the question of how to take such that the nonlinear matrix expression attains its maximal and minimal possible ranks.

1. Introduction

Throughout this paper denotes the set of all matrices over the complex field . denotes the identity matrix of order and is the matrix of all zero entries (if no confusion occurs, we will drop the subscript). For a matrix , and denote the conjugate transpose and the rank of the matrix , respectively. denotes a row block matrix consisting of and .

Let ; a generalized inverse of is a matrix which satisfies some of the following four Penrose equations [1]:

For a subset of the set , the set of matrices satisfying the equations from (1) is denoted by . A matrix in is called an -inverse of and is denoted by . For example, an matrix of the set is called a -inverse of and is denoted by . The unique -inverse of is denoted by , which is called the Moore-Penrose inverse of . For convenience, the symbols and stand for the two orthogonal projectors and . We refer the reader to [2–4] for basic results on generalized inverses.

In matrix theory and applications, there exists a nonlinear matrix expression that involves variable entries:

where is a given complex matrix and is a variable matrix. These nonlinear matrix expressions vary with respect to the choice of . One of the fundamental problems for (2) is to determine the maximal and minimal possible ranks of the matrix expression when is running over . Since the rank of matrix is an integer between 0 and the minimum of row and column numbers of the matrix [5], then the maximal and minimal ranks of can be attained for some .

The investigation of extremal ranks of matrix expressions has many direct motivations in matrix analysis. For example, a matrix expression of order is nonsingular if and only if the maximal rank of with respect to is ; two consistent matrix equations and have a common solution if and only if the minimal rank of the difference of their solutions is zero; a nonlinear matrix equation is consistent if and only if the minimal rank of with respect to is zero. From the definition of the -inverse of a matrix, we know that the solution of the nonlinear matrix equation is a -inverse of matrix ; that is, using the minimal rank of , we can find out the general expression of the -inverses of a matrix , which is a matrix such that the nonlinear matrix expression attains its minimal rank. In general, for any two matrix expressions and of the same size, there are and such that if and only if

These examples imply that the extremal ranks of matrix expressions have close links with many topics in matrix analysis and applications. Various statements on maximal and minimal ranks of matrix expressions are quite easy to understand for the people who know linear algebra. But the question now is how to give simple or closed forms for the extremal ranks of a matrix expression with respect to its variant matrices. The study on maximal and minimal ranks of matrix expression started in late 1980s. If want to know more about this question the reader can see [6–18].

The work in this paper includes two parts. First, in Section 2, we will consider how to choose a matrix , such that has the maximal possible rank. Second, in Section 3, we will determine the minimal rank of and present a general expression of the -inverses of matrix .

In order to find the extremal ranks of the nonlinear matrix expression , we need the following lemmas, which will be used in this paper.

Lemma 1 (see [19]). Let , where , , , , and are given matrices. Then for any variable matrices and , one has

where

Lemma 2 (see [20]). Let be a linear matrix expression over the complex field , where , , and are given; is a variant matrix. Then the maximal rank of with respect to is

the general expression of satisfying (7) is

where , , , and , and the matrix is chosen such that

Lemma 3 (see [21]). Let and . Then

where , , and .

Lemma 4 (see [22]). Let , , and . Then

where , , and .

2. The Maximal Rank of with respect to

Let be a given matrix; in this section, we will present the maximal rank of the nonlinear matrix expression , with respect to the variable matrix . The relative results are included in the following three lemmas.

Lemma 5. Let , and denotes the identity matrix of order . Then

Proof. By Lemma 3 with and , we have

where

Combining (13), (14) with (15), we have

The second equality holds as

Lemma 6. Let , and denotes the identity matrix of order . Then

The general expression of satisfying (18) is

where are chosen such that

Proof. By Lemma 2 with , , and , we have

From Lemmas 2 and 5, we have the general expression of satisfying (21) as

where

Combining (22), (23) with (24), we have , and

where are chosen such that

Lemma 7. Let . Then there exist some matrices , such that

Proof. Applying Lemma 4, we have

From (27) and (28), we have

By Lemma 1, we have

Combining (29) with (30), we get the result (27).

According to Lemmas 5, 6, and 7, we immediately obtain the following theorem.

Theorem 8. Let be a given matrix, and is a variant matrix. Then

In consequence,

(1)there always exists , such that is nonsingular;(2)the matrices satisfying (31) are given by and is the same as in (27).

Proof. First describe a special congruence transformation for a block matrix, which reduces the calculation of the maximal rank of :

By Lemma 1, we have

Combining (32) with (33), we have

On the other hand, from another special congruence transformation for a block matrix, we have

Combining formula (35) with Lemma 6, we have

That is, theres always exist , such that is nonsingular.

From the results in Lemmas 6 and 7, we obtain that there always exist , such that

3. The Minimal Rank of with respect to

In this section, we will present the minimal rank of the nonlinear matrix expression . Moreover, we will consider how to choose a matrix , such that has the minimal possible rank.

Theorem 9. Let be a given matrix, and is a variant matrix. Then

In consequence, there exists , such that the nonlinear matrix equation is consistent.

Proof. By formula (5) in Lemma 1, we have

where

Combining (39), (40) with (41), we have

Corollary 10. Let be a given matrix. Then the matrix satisfying the matrix equation is given by

where are two variant matrices.

Proof. Putting into yields

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

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

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