`Journal of Applied MathematicsVolumeΒ 2012, Article IDΒ 152805, 28 pageshttp://dx.doi.org/10.1155/2012/152805`
Research Article

## An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations

1School of Science, Sichuan University of Science and Engineering, Zigong 643000, China
2College of Management Science, Geomathematics Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu 610059, China

Received 14 November 2011; Revised 13 March 2012; Accepted 25 March 2012

Copyright Β© 2012 Feng Yin and Guang-Xin Huang. 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

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations , which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matrices and . When the matrix equations are consistent, for any initial generalized reflexive matrix pair , the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair to a given matrix pair in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair of a new corresponding generalized coupled Sylvester matrix equation pair , where . Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

#### 1. Introduction

In this paper, the following notations are used. Let denote the set of all real matrices. We denote by the superscript the transpose of a matrix. In matrix space , define inner product as for all , where denotes the trace of a matrix . represents the Frobenius norm of . represents the column space of . represents the vector operator, that is, for the matrix . stands for the Kronecker product of matrices and , denotes the block diagonal matrix with and and being the main diagonal elements orderly. denotes the n-order identity matrix.

Definition 1.1 (see [1, 2]). A matrix is said to be a generalized reflection matrix if satisfies that .

Definition 1.2 (see [1, 2]). Let and be two generalized reflection matrices. A matrix is called generalized reflexive (or generalized antireflexive) with respect to the matrix pair if . The set of all n-by-n generalized reflexive matrices with respect to matrix pair is denoted by .

The generalized reflexive and antireflexive matrices have many special properties and usefulness in engineering and scientific computations [1β6]. In particular, let , then a generalized reflexive matrix is called a reflexive matrix, which plays an important role in many areas and has been studied in [7β11]. Specially, let , then a reflexive matrix is called a generalized bisymmetric matrix, which has been studied in [12, 13]. Moreover, let , then a generalized reflexive matrix is the well-known centrosymmetric matrix, which has been widely and extensively studied in [14β17].

The generalized coupled Sylvester systems play a fundamental role in the various fields of engineering theory, particularly in control systems. The numerical solution of the generalized coupled Sylvester systems has been addressed in a large body of literature. KΓ₯gstrΓΆm and Westin [18] developed a generalized Schur method by applying the QZ algorithm to solve . Ding and Chen [19] presented an iterative least squares solutions of based on a hierarchical identification principle [20], in addition, by applying the hierarchical identification principle, KΔ±lΔ±Γ§man and Zhour [21] developed an iterative algorithm for obtaining the weighted least-squares solution. Recently, some finite iterative algorithms have also been developed to solve matrix equations. For more detail, we refer to [11, 13, 22β30]. Wang [31, 32] gave the bi(skew)symmetric and centrosymmetric solutions to the system of quaternion matrix equations . Wang [33] also solved a system of matrix equations over arbitrary regular rings with identity. Chang and Wang [34] gave the necessary and sufficient conditions for the existence of and the expressions for the symmetric solutions of the matrix equations , and . Ding and Chen [25] also presented the gradient-based iterative algorithms by applying the gradient search principle and the hierarchical identification principle for the general coupled matrix equations . Zhou et al. [35] proposed gradient-based iterative algorithms for solving the general coupled matrix equations with weighted least squares solutions. Wu et al. [36, 37] gave the finite iterative solutions to coupled Sylvester-conjugate matrix equations. Wu et al. [38] gave the finite iterative solutions to a class of complex matrix equations with conjugate and transpose of the unknowns. Jonsson and KΓ₯gstrΓΆm [39] proposed recursive block algorithms for solving the one-sided and coupled Sylvester matrix equations . Jonsson and KΓ₯gstrΓΆm [40] also proposed recursive block algorithms for the two-sided and generalized Sylvester and Lyapunov matrix equations. Dehghan and Hajarian [7, 8] gave the reflexive and generalized bisymmetric matrices solutions of the generalized coupled Sylvester matrix equations . Very recently, Dehghan and Hajarian [12] constructed an iterative algorithm to solve the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices. Huang et al. [13] present an iterative algorithm for the generalized coupled Sylvester matrix equations and its optimal approximation problem over generalized reflexive matrices solutions. In [30], the similar but different iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations and the optimal approximation problem over reflexive matrices. However, the generalized coupled Sylvester matrix equations and the optimal approximation over generalized reflexive matrices have not been solved.

In this paper, we will consider the following two problems.

Problem 1. Let , and be generalized reflection matrices. For given matrices ,β,β,β,β,β,β,β,β, find a pair of matrices , such that

Problem 2. When Problem 1 is consistent, let denote the set of the generalized reflexive solutions of Problem 1, that is, For a given matrix pair , find such that

The two-sided and generalized coupled Sylvester matrix equations (1.1) play a fundamental role in wide applications in several areas, such as stability theory, control theory, perturbation analysis, and some other fields of pure and applied mathematics. In addition, as special type of generalized coupled Sylvester matrix equations (1.1), the generalized Sylvester matrix equation arises in computing the deflating subspace of descriptor linear systems [18]. Wu et al. [36] presented some examples to show a motivation for studying (1.1). Problem 2 occurs frequently in experiment design, see for instance [41].

This paper is organized as follows. In Section 2, we will solve Problem 1 by constructing an iterative algorithm, that is, if Problem 1 is consistent, then for an arbitrary initial matrix pair , we can obtain a solution pair of Problem 1 within finite iterative steps in the absence of round-off errors. Let and , where , are arbitrary matrices, or more especially, let and , we can obtain the least Frobenius norm solutions of Problem 1. Then, in Section 3, we give the optimal approximate solution pair of Problem 2 by finding the least Frobenius norm generalized reflexive solution pair of the corresponding generalized coupled Sylvester matrix equations. In Section 4, several numerical examples are given to illustrate the application of our method. At last, some conclusions are drawn in Section 5.

#### 2. An Iterative Algorithm for Solving Problem 1

In this section, we will first introduce an iterative algorithm to solve Problem 1, then prove that it is convergent. Then, we will give the least-norm generalized reflexive solutions of Problem 1 when an appropriate initial iterative matrix pair is chosen.

For the purpose of simplification, we introduce the following operators:

Algorithm 2.1. We have the following steps.

Step 1. Input matrices ,β,β,β,β, ,β,β,β,β, and four generalized reflection matrices ,β, .

Step 2. Choose two arbitrary matrices , . Compute

Step 3. If , then stop and is the solution of the generalized coupled Sylvester matrix equation (1.1); else if , but and , then stop and the generalized coupled Sylvester matrix equations (1.1) are not consistent over generalized reflexive matrices; else .

Step 4. Compute

Step 5. Go to Step 3.
Obviously, it can be seen that , , where .

Lemma 2.2. For the sequences , , and generated by Algorithm 2.1, and , we have

The proof of Lemma 2.2 is presented in Appendix A.

Lemma 2.3. Suppose is an arbitrary solution pair of Problem 1, then for any initial generalized reflexive matrix pair , we have where the sequences ,, , , and are generated by Algorithm 2.1.

The proof of Lemma 2.3 is presented in Appendix B.

Remark 2.4. If there exist, a positive number such that and but , then by Lemma 2.3, we have that the generalized coupled Sylvester matrix equations (1.1) are not consistent over generalized reflexive matrices.

Theorem 2.5. Suppose that Problem 1 is consistent, then for an arbitrary initial matrix pair , a generalized reflexive solution pair of Problem 1 can be obtained with finite iteration steps in the absence of round-off errors.

Proof. If , by Lemma 2.3, we have , then we can compute by Algorithm 2.1.
By Lemma 2.2, we have
It can be seen that the set of is an orthogonal basis of the matrix subspace which implies that , that is, is a solution pair of Problem 1. This completes the proof.

To show the least Frobenius norm generalized reflexive solutions of Problem 1, we first introduce the following result.

Lemma 2.6 (see [42, Lemma 2.4]). Suppose that the consistent system of linear equation has a solution , then is a unique least Frobenius norm solution of the system of linear equation.

By Lemma 2.6, the following result can be obtained.

Theorem 2.7. Suppose that Problem 1 is consistent. If we choose the initial iterative matrices and β, where are arbitrary matrices, especially, and , then the solution pair generated by Algorithm 2.1 is the unique least Frobenius norm generalized reflexive solutions of Problem 1.

Proof. We know the solvability of the generalized coupled Sylvester matrix equations (1.1) over generalized reflexive matrices is equivalent to the following matrix equations: Then, the system of matrix equations (2.8) is equivalent to Let and β , where are arbitrary matrices, then Furthermore, we can see that all generated by Algorithm 2.1 satisfy by Lemma 2.6, we know that is the least Frobenius norm generalized reflexive solution pair of the system of linear equations (2.9). Since vector operator is isomorphic, is the unique least Frobenius norm generalized reflexive solution pair of the system of matrix equations (2.8), then is the unique least Frobenius norm generalized reflexive solution pair of Problem 1.

#### 3. The Solution of Problem 2

In this section, we will show that the optimal approximate solutions of Problem 2 for a given generalized reflexive matrix pair can be derived by finding the least Frobenius norm generalized reflexive solutions of the corresponding generalized coupled Sylvester matrix equations.

When Problem 1 is consistent, the set of generalized reflexive solutions of Problem 1 denoted by is not empty. For a given matrix pair , we have Set , then Problem 2 is equivalent to that of finding the least Frobenius norm generalized reflexive solutions pair of the corresponding generalized coupled Sylvester matrix equations By using Algorithm 2.1, let initial iteration matrix and , or more especially, let and , then we can get the least Frobenius norm generalized reflexive solution pair of (3.2). Thus, the generalized reflexive solution pair of the problem 2 can be represented as .

#### 4. Numerical Experiments

In this section, we will show several numerical examples to illustrate our results. All the tests are performed by MATLAB 7.8.

Example 4.1. Consider the generalized reflexive solutions of the generalized coupled Sylvester matrix equations , where Let be generalized reflection matrices.
We will find the generalized reflexive solutions of the matrix equations by using Algorithm 2.1. It can be verified that the matrix equations are consistent over generalized reflexive matrices and the solutions are Because of the influence of the error of calculation, the residual is usually unequal to zero in the process of the iteration, where . For any chosen positive number ; however, small enough, for example, , whenever , stop the iteration, and are regarded to be generalized reflexive solutions of the matrix equations . Choose an initially iterative matrix pair , such as By Algorithm 2.1, we have So we obtain the generalized reflexive solutions of the matrix equations . The relative error of the solutions and the residual are shown in Figure 1, where the relative error and the residual .
Let by Algorithm 2.1, we have The relative error of the solutions and the residual are shown in Figure 2.

Figure 1: The relative error of the solutions and the residual for Example 4.1 with .
Figure 2: The relative error of the solutions and the residual for Example 4.1 with .

Example 4.2. Consider the unique least-norm generalized reflexive solutions of the matrix equations in Example 4.1. Let By using Algorithm 2.1, we have the least-norm generalized reflexive solutions of the matrix equations as follows:
The relative error of the solutions and the residual are shown in Figure 3.

Figure 3: The relative error of the solutions and the residual for Example 4.2.

Example 4.3. Let denote the set of all generalized reflexive solutions of the matrix equations in Example 4.1. For a given generalized reflexive matrices we will find , such that that is, find the optimal approximate generalized reflexive solution pair to the matrix pair in in Frobenius norm.
Let , by the method mentioned in Section 3, we can obtain the least-norm generalized reflexive solution pair of the matrix equations by choosing the initial iteration matrices and , then by Algorithm 2.1, we have thatand the optimal approximate generalized reflexive solutions to the matrix pair in Frobenius norm are The relative error of the solutions and the residual are shown in Figure 4, where the relative error and the residual .

Figure 4: The relative error of the solutions and the residual for Example 4.3.

#### 5. Conclusions

In this paper, an efficient iterative algorithm is presented to solve the generalized coupled Sylvester matrix equations over generalized reflexive matrix pair . When the matrix equations are consistent over generalized reflexive matrices and , for any generalized reflexive initial iterative matrix pair , the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors. Let initial matrices and , where are arbitrary matrices, especially, let and , the unique least-norm generalized reflexive solutions of the matrix equations can be derived. Furthermore, the optimal approximate solutions of for a given generalized reflexive matrix pair can be derived by finding the least-norm generalized reflexive solutions of two new corresponding generalized coupled Sylvester matrix equations. Finally, several numerical examples are given to illustrate that our iterative algorithm is quite effective.

The results presented in this paper generalize some previous results [7, 12, 13, 30]. When , , and , then our results reduce to those in [7]. When , and , the results in this paper reduce to those in [12]. When , and , then the results in this paper reduce to those in [13]. When and , then the results in this paper reduce to those in [30].

#### A. The Proof of Lemma 2.2

Since , , and for all , we only need to prove that We prove the conclusion by induction, and two steps are required.

Step 1. We will show that
To prove this conclusion, we also use induction.
For , by Algorithm 2.1, we have that
Assume that (A.2) holds for , that is, .
When , we have that Hence, (A.2) holds for . Therefore, (A.2) holds by the principle of induction.

Step 2. We show that
When , (A.5) holds.
Assume that then we show that
In fact, we have that