Journal of Applied Mathematics

Volume 2012 (2012), Article ID 398085, 14 pages

doi:10.1155/2012/398085

Research Article

On the Hermitian -Conjugate Solution of a System of Matrix Equations

Chang-Zhou Dong,¹ Qing-Wen Wang,² and Yu-Ping Zhang³

¹School of Mathematics and Science, Shijiazhuang University of Economics, Shijiazhuang, Hebei 050031, China

²Department of Mathematics, Shanghai University, Shanghai, Shanghai 200444, China

³Department of Mathematics, Ordnance Engineering College, Shijiazhuang, Hebei 050003, China

Received 3 October 2012; Accepted 26 November 2012

Academic Editor: Yang Zhang

Copyright © 2012 Chang-Zhou Dong et al. 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.

Let be an by nontrivial real symmetric involution matrix, that is, . An complex matrix is termed -conjugate if , where denotes the conjugate of . We give necessary and sufficient conditions for the existence of the Hermitian -conjugate solution to the system of complex matrix equations and present an expression of the Hermitian -conjugate solution to this system when the solvability conditions are satisfied. In addition, the solution to an optimal approximation problem is obtained. Furthermore, the least squares Hermitian -conjugate solution with the least norm to this system mentioned above is considered. The representation of such solution is also derived. Finally, an algorithm and numerical examples are given.

1. Introduction

Throughout, we denote the complex matrix space by , the real matrix space by , and the set of all matrices in with rank by . The symbols , and stand for the identity matrix with the appropriate size, the conjugate, the transpose, the conjugate transpose, the Moore-Penrose generalized inverse, and the Frobenius norm of , respectively. We use to denote the backward matrix having the elements along the southwest diagonal and with the remaining elements being zeros.

Recall that an complex matrix is centrohermitian if . Centrohermitian matrices and related matrices, such as k-Hermitian matrices, Hermitian Toeplitz matrices, and generalized centrohermitian matrices, appear in digital signal processing and others areas (see, [1–4]). As a generalization of a centrohermitian matrix and related matrices, Trench [5] gave the definition of -conjugate matrix. A matrix is -conjugate if , where is a nontrivial real symmetric involution matrix, that is, and . At the same time, Trench studied the linear equation for -conjugate matrices in [5], where are known column vectors.

Investigating the matrix equation

with the unknown matrix being symmetric, reflexive, Hermitian-generalized Hamiltonian, and repositive definite is a very active research topic [6–14]. As a generalization of (1.1), the classical system of matrix equations

has attracted many author’s attention. For instance, [15] gave the necessary and sufficient conditions for the consistency of (1.2), [16, 17] derived an expression for the general solution by using singular value decomposition of a matrix and generalized inverses of matrices, respectively. Moreover, many results have been obtained about the system (1.2) with various constraints, such as bisymmetric, Hermitian, positive semidefinite, reflexive, and generalized reflexive solutions (see, [18–28]). To our knowledge, so far there has been little investigation of the Hermitian -conjugate solution to (1.2).

Motivated by the work mentioned above, we investigate Hermitian -conjugate solutions to (1.2). We also consider the optimal approximation problem

where is a given matrix in and the set of all Hermitian -conjugate solutions to (1.2). In many cases the system (1.2) has not Hermitian -conjugate solution. Hence, we need to further study its least squares solution, which can be described as follows: Let denote the set of all Hermitian -conjugate matrices in :

Find such that

In Section 2, we present necessary and sufficient conditions for the existence of the Hermitian -conjugate solution to (1.2) and give an expression of this solution when the solvability conditions are met. In Section 3, we derive an optimal approximation solution to (1.3). In Section 4, we provide the least squares Hermitian -conjugate solution to (1.5). In Section 5, we give an algorithm and a numerical example to illustrate our results.

2. -Conjugate Hermitian Solution to (1.2)

In this section, we establish the solvability conditions and the general expression for the Hermitian -conjugate solution to (1.2).

We denote and the set of all -conjugate matrices and Hermitian -conjugate matrices, respectively, that is,

where is nontrivial real symmetric involution matrix.

Chang et al. in [29] mentioned that for nontrivial symmetric involution matrix , there exist positive integer and real orthogonal matrix such that

where . By (2.2),

Throughout this paper, we always assume that the nontrivial symmetric involution matrix is fixed which is given by (2.2) and (2.3). Now, we give a criterion of judging a matrix is -conjugate Hermitian matrix.

Theorem 2.1. A matrix if and only if there exists a symmetric matrix such that , where

with being the same as (2.2).

Proof. If , then . By (2.2),

which is equivalent to

Suppose that

Substituting (2.7) into (2.6), we obtain

Hence, are real matrices. If we denote , then by (2.7)

Let , and

Then, can be expressed as , where is unitary matrix and is a real matrix. By

we obtain .

Conversely, if there exists a symmetric matrix such that , then it follows from (2.3) that

that is, .

Theorem 2.1 implies that an arbitrary complex Hermitian -conjugate matrix is equivalent to a real symmetric matrix.

Lemma 2.2. For any matrix , where

Proof. For any matrix , it is obvious that , where are defined as (2.13). Now, we prove that the decomposition is unique. If there exist such that , then

It follows from are real matrix that

Hence, holds, where are defined as (2.13).

By Theorem 2.1, for , we may assume that

where is defined as (2.4) and is a symmetric matrix.

Suppose that , and , where

Then, system (1.2) can be reduced into

which implies that

Let

Then, system (1.2) has a solution in if and only if the real system

has a symmetric solution in .

Lemma 2.3 (Theorem  1 in [7]). Let . The of matrix is as follows

where and are orthogonal matrices, . Then, (1.1) has a symmetric solution if and only if

In that case, it has the general solution

where is an arbitrary symmetric matrix.

By Lemma 2.3, we have the following theorem.

Theorem 2.4. Given , and . Let , be defined in (2.17), (2.20), respectively. Assume that the of is as follows

where and are orthogonal matrices, . Then, system (1.2) has a solution in if and only if

In that case, it has the general solution

where is an arbitrary symmetric matrix.

3. The Solution of Optimal Approximation Problem (1.3)

When the set of all Hermitian -conjugate solution to (1.2) is nonempty, it is easy to verify is a closed set. Therefore, the optimal approximation problem (1.3) has a unique solution by [30].

Theorem 3.1. Given , and . Assume is nonempty, then the optimal approximation problem (1.3) has a unique solution and

Proof. Since is nonempty, has the form of (2.27). By Lemma 2.2, can be written as

where

According to (3.2) and the unitary invariance of Frobenius norm

We get

Then, is consistent if and only if there exists such that

For the orthogonal matrix

Therefore,

is equivalent to

Substituting (3.9) into (2.27), we obtain (3.1).

4. The Solution of Problem (1.5)

In this section, we give the explicit expression of the solution to (1.5).

Theorem 4.1. Given ,  and  . Let ,  and   be defined in (2.17), (2.20), respectively. Assume that the of is as (2.25) and system (1.2) has not a solution in . Then, can be expressed as

where is an arbitrary symmetric matrix.

Proof. It yields from (2.17)–(2.21) and (2.25) that

Assume that

Then, we have

Hence,

is solvable if and only if there exist such that

It follows from (4.6) that

Substituting (4.7) into (4.3) and then into (2.16), we can get that the form of elements in is (4.1).

Theorem 4.2. Assume that the notations and conditions are the same as Theorem 4.1. Then,

if and only if

Proof. In Theorem 4.1, it implies from (4.1) that is equivalent to has the expression (4.1) with . Hence, (4.9) holds.

5. An Algorithm and Numerical Example

Base on the main results of this paper, we in this section propose an algorithm for finding the solution of the approximation problem (1.3) and the least squares problem with least norm (1.5). All the tests are performed by MATLAB 6.5 which has a machine precision of around .

Algorithm 5.1. Input .

Compute by (2.17) and (2.20).

Compute the singular value decomposition of with the form of (2.25).

If (2.26) holds, then input and compute the solution of problem (1.3) according (3.1), else compute the solution to problem (1.5) by (4.9).

To show our algorithm is feasible, we give two numerical example. Let an nontrivial symmetric involution be

We obtain in (2.2) by using the spectral decomposition of , then by (2.4)

Example 5.2. Suppose , and

We can verify that (2.26) holds. Hence, system (1.2) has an Hermitian -conjugate solution. Given

Applying Algorithm 5.1, we obtain the following:

Example 5.2 illustrates that we can solve the optimal approximation problem with Algorithm 5.1 when system (1.2) have Hermitian -conjugate solutions.

Example 5.3. Let be the same as Example 5.2, and let in Example 5.2 be changed into

We can verify that (2.26) does not hold. By Algorithm 5.1, we get

Example 5.3 demonstrates that we can get the least squares solution with Algorithm 5.1 when system (1.2) has not Hermitian -conjugate solutions.

Acknowledgments

This research was supported by the Grants from the Key Project of Scientific Research Innovation Foundation of Shanghai Municipal Education Commission (13ZZ080), the National Natural Science Foundation of China (11171205), the Natural Science Foundation of Shanghai (11ZR1412500), the Ph.D. Programs Foundation of Ministry of Education of China (20093108110001), the Discipline Project at the corresponding level of Shanghai (A. 13-0101-12-005), Shanghai Leading Academic Discipline Project (J50101), the Natural Science Foundation of Hebei province (A2012403013), and the Natural Science Foundation of Hebei province (A2012205028). The authors are grateful to the anonymous referees for their helpful comments and constructive suggestions.

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