Journal of Applied Mathematics

Volume 2013 (2013), Article ID 593549, 11 pages

http://dx.doi.org/10.1155/2013/593549

Research Article

Solving Optimization Problems on Hermitian Matrix Functions with Applications

Xiang Zhang and Shu-Wen Xiang

Department of Computer Science and Information, Guizhou University, Guiyang 550025, China

Received 17 October 2012; Accepted 20 March 2013

Academic Editor: K. Sivakumar

Copyright © 2013 Xiang Zhang and Shu-Wen Xiang. 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 extremal inertias and ranks of the matrix expressions , where , and are known matrices and and are the solutions to the matrix equations , , and , respectively. As applications, we present necessary and sufficient condition for the previous matrix function to be positive (negative), non-negative (positive) definite or nonsingular. We also characterize the relations between the Hermitian part of the solutions of the above-mentioned matrix equations. Furthermore, we establish necessary and sufficient conditions for the solvability of the system of matrix equations , , , and , and give an expression of the general solution to the above-mentioned system when it is solvable.

1. Introduction

Throughout, we denote the field of complex numbers by , the set of all matrices over by , and the set of all Hermitian matrices by . The symbols and stand for the conjugate transpose, the column space of a complex matrix respectively. denotes the identity matrix. The Moore-Penrose inverse [1] of , is the unique solution to the four matrix equations:

Moreover, and stand for the projectors induced by . It is well known that the eigenvalues of a Hermitian matrix are real, and the inertia of is defined to be the triplet

where , , and stand for the numbers of positive, negative, and zero eigenvalues of , respectively. The symbols and are called the positive index and the negative index of inertia, respectively. For two Hermitian matrices and of the same sizes, we say () in the Löwner partial ordering if is positive (negative) semidefinite. The Hermitian part of is defined as . We will say that is Re-nnd (Re-nonnegative semidefinite) if is Re-pd (Re-positive definite) if , and is Re-ns if is nonsingular.

It is well known that investigation on the solvability conditions and the general solution to linear matrix equations is very active (e.g., [2–9]). In 1999, Braden [10] gave the general solution to

In 2007, Djordjević [11] considered the explicit solution to (3) for linear bounded operators on Hilbert spaces. Moreover, Cao [12] investigated the general explicit solution to

Xu et al. [13] obtained the general expression of the solution of operator equation (4). In 2012, Wang and He [14] studied some necessary and sufficient conditions for the consistence of the matrix equation

and presented an expression of the general solution to (5).

Note that (5) is a special case of the following system:

To our knowledge, there has been little information about (6). One goal of this paper is to give some necessary and sufficient conditions for the solvability of the system of matrix (6) and present an expression of the general solution to system (6) when it is solvable.

In order to find necessary and sufficient conditions for the solvability of the system of matrix equations (6), we need to consider the extremal ranks and inertias of (10) subject to (13) and (11).

There have been many papers to discuss the extremal ranks and inertias of the following Hermitian expressions:

Tian has contributed much in this field. One of his works [15] considered the extremal ranks and inertias of (7). He and Wang [16] derived the extremal ranks and inertias of (7) subject to , . Liu and Tian [17] studied the extremal ranks and inertias of (8). Chu et al. [18] and Liu and Tian [19] derived the extremal ranks and inertias of (9). Zhang et al. [20] presented the extremal ranks and inertias of (9), where and are Hermitian solutions of

respectively. He and Wang [16] derived the extremal ranks and inertias of (10). We consider the extremal ranks and inertias of (10) subject to (11) and

which is not only the generalization of the above matrix functions, but also can be used to investigate the solvability conditions for the existence of the general solution to the system (6). Moreover, it can be applied to characterize the relations between Hermitian part of the solutions of (11) and (13).

The remainder of this paper is organized as follows. In Section 2, we consider the extremal ranks and inertias of (10) subject to (11) and (13). In Section 3, we characterize the relations between the Hermitian part of the solution to (11) and (13). In Section 4, we establish the solvability conditions for the existence of a solution to (6) and obtain an expression of the general solution to (6).

2. Extremal Ranks and Inertias of Hermitian Matrix Function (10) with Some Restrictions

In this section, we consider formulas for the extremal ranks and inertias of (10) subject to (11) and (13). We begin with the following Lemmas.

Lemma 1 (see [21]). (a) Let , , , and be given. Then the following statements are equivalent:

(1)system (13) is consistent,(2)

(3)

In this case, the general solution can be written as

where is arbitrary.

(b) Let and be given. Then the following statements are equivalent:

(1)equation (11) is consistent,(2)

(3)

In this case, the general solution can be written as

where is arbitrary.

Lemma 2 ([22, Lemma 1.5, Theorem 2.3]). Let , , and , and denote that

Then one has the following

(a)the following equalities hold

(b)if , then . Thus if and only if and ,(c)

Lemma 3 (see [23]). Let , , and . Then they satisfy the following rank equalities:

(a), (b), (c), (d), (e), (f),

Lemma 4 (see [15]). Let , , , , and be given, and be nonsingular. Then one has the following

(1), (2), (3), (4).

Lemma 5 (see [22, Lemma  1.4]). Let be a set consisting of (square) matrices over , and let be a set consisting of (square) matrices over . Then Then one has the following

(a) has a nonsingular matrix if and only if ; (b)any is nonsingular if and only if ; (c) if and only if ; (d) if and only if ;(e) has a matrix if and only if ;(f)any satisfies if and only if ); (g) has a matrix if and only if ; (h)any satisfies if and only if .

Lemma 6 (see [16]). Let , where , , , and are given with appropriate sizes, and denote that

Then one has the following:

(1)the maximal rank of is

(2)the minimal rank of is

(3)the maximal inertia of is

(4)the minimal inertias of is

where

Now we present the main theorem of this section.

Theorem 7. Let , , , , , , , ,   , and be given, and suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by and (11) by . Put

Then one has the following:

(a)the maximal rank of (10) subject to (13) and (11) is

(b)the minimal rank of (10) subject to (13) and (11) is

(c)the maximal inertia of (10) subject to (13) and (11) is

(d)the minimal inertia of (10) subject to (13) and (11) is

Proof. By Lemma 1, the general solutions to (13) and (11) can be written as

where and are arbitrary matrices with appropriate sizes. Put

Substituting (36) into (10) yields

Clearly is Hermitian. It follows from Lemma 6 that

where

Now, we simplify the ranks and inertias of block matrices in (38)–(41).

By Lemma 4, block Gaussian elimination, and noting that

we have the following:

By , we obtain

By Lemma 2, we can get the following:

Substituting (44)-(47) into (38) and (41) yields (31)–(34), respectively.

Corollary 8. Let , , , , , , , , , , and , () be as in Theorem 7, and suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by and (11) by . Then, one has the following:

(a)there exist and such that if and only if

(b)there exist and such that if and only if

(c)there exist and such that if and only if

(d)there exist and such that if and only if

(e) for all and if and only if

(f) for all and if and only if

(g) for all and if and only if

(h) for all and if and only if

(i)there exist and such that is nonsingular if and only if

3. Relations between the Hermitian Part of the Solutions to (13) and (11)

Now we consider the extremal ranks and inertias of the difference between the Hermitian part of the solutions to (13) and (11).

Theorem 9. Let , , , , , and , be given. Suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by and (11) by . Put

Then one has the following:

Proof. By letting , , , and in Theorem 7, we can get the results.

Corollary 10. Let , , , , , , and , () be as in Theorem 9, and suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by and (11) by . Then, one has the following:

(a)there exist and such that if and only if

(b)there exist and such that if and only if

(c)there exist and such that if and only if

(d)there exist and such that if and only if

(e) for all and if and only if

(f) for all and if and only if

(g) for all and if and only if

(h) for all and if and only if

(i)there exist and such that is nonsingular if and only if

4. The Solvability Conditions and the General Solution to System (6)

We now turn our attention to (6). We in this section use Theorem 9 to give some necessary and sufficient conditions for the existence of a solution to (6) and present an expression of the general solution to (6). We begin with a lemma which is used in the latter part of this section.

Lemma 11 (see [14]). Let , , , and be given. Let , , , , , and . Then the following statements are equivalent:

(1)equation (5) is consistent,(2)

(3)

In this case, the general solution of (5) can be expressed as

where , , , , , and are arbitrary matrices over with appropriate sizes.

Now we give the main theorem of this section.

Theorem 12. Let , , (), , and , () be given. Set

Then the following statements are equivalent:

(1)system (6) is consistent,(2)the equalities in (14) and (17) hold, and

(3)the equalities in (15) and (18) hold, and

In this case, the general solution of system (6) can be expressed as

where

where , , , , , and are arbitrary matrices over with appropriate sizes.

Proof. (2) (3): Applying Lemma 3 and Lemma 11 gives

By a similar approach, we can obtain that

(1) (2): We separate the four equations in system (6) into three groups:

By Lemma 1, we obtain that system (80) is solvable if and only if (14), (81) is consistent if and only if (17). The general solutions to system (80) and (81) can be expressed as (16) and (19), respectively. Substituting (16) and (19) into (82) yields

Hence, the system (5) is consistent if and only if (80), (81), and (83) are consistent, respectively. It follows from Lemma 11 that (83) is solvable if and only if

We know by Lemma 11 that the general solution of (83) can be expressed as (77).

In Theorem 12, let and vanish. Then we can obtain the general solution to the following system:

Corollary 13. Let , , , , , , , and be given. Set

Then the following statements are equivalent:

(1)system (85) is consistent(2)

(3)

In this case, the general solution of system (6) can be expressed as

where

where , , , , , and are arbitrary matrices over with appropriate sizes.

Acknowledgments

The authors would like to thank Dr. Mohamed, Dr. Sivakumar, and a referee very much for their valuable suggestions and comments, which resulted in a great improvement of the original paper. This research was supported by the Grants from the National Natural Science Foundation of China (NSFC (11161008)) and Doctoral Program Fund of Ministry of Education of P.R.China (20115201110002).

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