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

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations AXB − CYD,EXF − GYH M,N , which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matrices X and Y . When the matrix equations are consistent, for any initial generalized reflexive matrix pair X1, Y1 , 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 ̂ X, ̂ Y to a given matrix pair X0, Y0 in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair ̃ X∗, ̃ Y ∗ of a new corresponding generalized coupled Sylvester matrix equation pair A ̃ XB − C ̃ YD,E ̃ XF −G ̃ YH ̃ M, ̃ N , where


Introduction
In this paper, the following notations are used.Let R m×n denote the set of all m × n real matrices.We denote by the superscript T the transpose of a matrix.In matrix space R m×n , define inner product as A, B tr B T A for all A, B ∈ R m×n , where tr A denotes the trace of a matrix A. A represents the Frobenius norm of A. R A represents the column space of A. vec • represents the vector operator, that is, vec A a T 1 , a T 2 , . . ., a T n T ∈ R mn for the matrix A a 1 , a 2 , . . ., a n ∈ R m×n , a i ∈ R m , i 1, 2, . . ., n.A ⊗ B stands for the Kronecker product of matrices A and B, diag A, B denotes the block diagonal matrix with A and B and being the main diagonal elements orderly.I n denotes the n-order identity matrix.
Definition 1.1 see 1, 2 .A matrix P ∈ R n×n is said to be a generalized reflection matrix if P satisfies that P T P, P 2 I.
Definition 1.2 see 1, 2 .Let P ∈ R n×n and Q ∈ R n×n be two generalized reflection matrices.A matrix A ∈ R n×n is called generalized reflexive or generalized antireflexive with respect to the matrix pair P, Q if PAQ A or PAQ −A .The set of all n-by-n generalized reflexive matrices with respect to matrix pair P, Q is denoted by R n×n r P, Q .
The generalized reflexive and antireflexive matrices have many special properties and usefulness in engineering and scientific computations 1-6 .In particular, let P Q, then a generalized reflexive matrix is called a reflexive matrix, which plays an important role in many areas and has been studied in 7-11 .Specially, let X T X, then a reflexive matrix X is called a generalized bisymmetric matrix, which has been studied in 12, 13 .Moreover, let P Q J n , then a generalized reflexive matrix is the well-known centrosymmetric matrix, which has been widely and extensively studied in 14-17 .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.Kågstr öm and Westin 18 developed a generalized Schur method by applying the QZ algorithm to solve AXB − CY D, EXF − GY H M, N .Ding and Chen 19 presented an iterative least squares solutions of AXB − CY D, EXF − GY H M, N based on a hierarchical identification principle 20 , in addition, by applying the hierarchical identification principle, Kılıc ¸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, 22-30 .Wang 31,32 gave the bi skew symmetric and centrosymmetric solutions to the system of quaternion matrix equations A 1 X C 1 , A 3 XB 3 C 3 .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 AX Y A C, AXA T BY B T C, and A T XA, B T XB C, D .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 M, N and the optimal approximation problem over reflexive matrices.However, the generalized coupled Sylvester matrix equations AXB − CY D, EXF − GY H M, N and the optimal approximation over generalized reflexive matrices have not been solved.
In this paper, we will consider the following two problems.
, and S ∈ R t×t be generalized reflection matrices.For given matrices 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.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 R, S , we can obtain a solution pair Y * , Z * of Problem 1 within finite iterative steps in the absence of round-off errors.Let where K ∈ R p×q , L ∈ R k×l are arbitrary matrices, or more especially, let X 1 0 and Y 1 0, 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.

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:
, and four generalized reflection matrices Step 2. Choose two arbitrary matrices

2.2
Step 3. If R k 0, then stop and X k , Y k is the solution of the generalized coupled Sylvester matrix equation 1.1 ; else if R k / 0, but U k 0 and V k 0, then stop and the generalized coupled Sylvester matrix equations 1.1 are not consistent over generalized reflexive matrices; else k : k 1.
Obviously, it can be seen that The proof of Lemma 2.2 is presented in Appendix A.
Lemma 2.3.Suppose X * , Y * is an arbitrary solution pair of Problem 1, then for any initial generalized reflexive matrix pair X 1 , Y 1 , we have where the sequences {X i },{Y i }, {U i }, {V i }, and {R i } 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 k such that U k 0 and V k 0 but R k / 0, 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

S , a generalized reflexive solution pair of Problem 1 can be obtained with finite iteration steps in the absence of round-off errors.
Proof.If R i / 0, i 1, 2, . . ., pq st, by Lemma 2.3, we have U i / 0, V i / 0, i 1, 2, . . ., pq st, then we can compute X pq st 1 , Y pq st 1 by Algorithm 2.1.
By Lemma 2.2, we have

2.6
It can be seen that the set of R 1 , R 2 , . . ., R pq st is an orthogonal basis of the matrix subspace , S 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 Ax b has a solution x * ∈ R A T , then x * is a unique least Frobenius norm solution of the system of linear equation.
By Lemma 2.6, the following result can be obtained.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:

2.8
Then, the system of matrix equations 2.8 is equivalent to

2.10
Furthermore, we can see that all X k , Y k generated by Algorithm 2.1 satisfy by Lemma 2.6, we know that X * , Y * is the least Frobenius norm generalized reflexive solution pair of the system of linear equations 2.9 .Since vector operator is isomorphic, X * , Y * is the unique least Frobenius norm generalized reflexive solution pair of the system of matrix equations 2.8 , then X * , Y * is the unique least Frobenius norm generalized reflexive solution pair of Problem 1.

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.

Journal of Applied Mathematics
When Problem 1 is consistent, the set of generalized reflexive solutions of Problem 1 denoted by S E is not empty.For a given matrix pair is equivalent to that of finding the least Frobenius norm generalized reflexive solutions pair X * , Y * of the corresponding generalized coupled Sylvester matrix equations

3.2
By using Algorithm 2.1, let initial iteration matrix , S , then we can get the least Frobenius norm generalized reflexive solution pair X * , Y * of 3.2 .Thus, the generalized reflexive solution pair of the problem 2 can be represented as X, Y X * X 0 , Y * Y 0 .

Numerical Experiments
In this section, we will show several numerical examples to illustrate our results.All the tests are performed by MATLAB 7.8.
be generalized reflection matrices.
We will find the generalized reflexive solutions of the matrix equations AXB − CY D M, EXY − GY H N 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 R i is usually unequal to zero in the process of the iteration, where i 1, 2, . ... For any chosen positive number ε; however, small enough, for example, ε 1.0000e − 010, whenever R k < ε, stop the iteration, X k and Y k are regarded to be generalized reflexive solutions of the matrix equations AXB − CY D M, EXY −GY H N. Choose an initially iterative matrix pair By Algorithm 2.1, we have −2.0000 9.0000 2.0000 5.0000 3.0000 1.0000 11.0000 −1.0000 7.0000 3.0000 −7.0000 3.0000 11.0000 1.0000 3.0000 −1.0000 −2.0000 5.0000 2.0000 9.0000 14.0000 16.0000 −1.0000 3.0000 4.0000 9.0000 7.0000 0 9.0000 7.0000 −3.0000 −8.0000 −8.0000 3.0000 8.0000 3.0000 4.0000 1.0000 14.0000 16.0000

4.9
The relative error of the solutions and the residual are shown in Figure 3.

4.13
The relative error of the solutions and the residual are shown in Figure 4, where the relative error REk

Conclusions
In  iterative steps in the absence of round-off errors.Let initial matrices R, S , the unique least-norm generalized reflexive solutions of the matrix equations can be derived.Furthermore, the optimal approximate solutions of AXB − CY D M, EXY − GY H N for a given generalized reflexive matrix pair X 0 , Y 0 ∈ R m×n r P, Q × R s×t r R, S 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 B I, C I, F I, G I, P Q, and R S, then our results reduce to those in 7 .When P Q, R S, X T X, and Y T Y , the results in this paper reduce to those in 12 .When B I, C I, F I, and G I, then the results in this paper reduce to those in 13 .When P Q and R S, then the results in this paper reduce to those in 30 .

A. The Proof of Lemma 2.2
Since tr R T i R j tr R T j R i , tr U T i U j tr U T j U i , and tr V T i V j tr V T j V i for all i, j 1, 2, . . ., s, we only need to prove that tr R T i R j 0, tr We prove the conclusion by induction, and two steps are required.
Step 1.We will show that tr To prove this conclusion, we also use induction.
For i 1, by Algorithm 2.1, we have that Hence, A.2 holds for i k.Therefore, A.2 holds by the principle of induction.

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Step 2. We show that tr In fact, we have that tr From the above results, we have tr R T i 1 R j 1 0, j 1, 2, . . ., s − 1, and A.9 By the principle of induction, A.5 holds.Noting that A.1 is implied in Steps 1 and 2 by the principle of induction.This completes the proof.

B. The Proof of Lemma 2.3
We proof the conclusion by induction.
For i 1, we have that tr

B.1
Assume that 2.5 holds for i k.When i k 1, by Algorithm 2.1, we have that tr B.2 then the solution pair Y * , Z * generated by Algorithm 2.1 is the unique least Frobenius norm generalized reflexive solutions of Problem 1.

Example 4 . 1 .
Consider the generalized reflexive solutions of the generalized coupled Sylvester matrix equations AXB − CY D M, EXY − GY H N, where

Example 4 . 2 .
Consider the unique least-norm generalized reflexive solutions of the matrix equations in Example 4.1.Let K

Figure 1 :
Figure 1: The relative error of the solutions and the residual for Example 4.1 with X 1 / 0, Y 1 / 0.

Figure 2 :
Figure 2: The relative error of the solutions and the residual for Example 4.1 with X 1 0, Y 1 0.
by the method mentioned in Section 3, we can obtain the least-norm generalized reflexive solution Journal of Applied Mathematics pair X * , Y * of the matrix equations A XB C Y D M, E XF G Y H N by choosing the initial iteration matrices X 1 0 and Y 1 0, then by Algorithm 2.1, we have that

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

Figure 4 :
Figure 4: The relative error of the solutions and the residual for Example 4.3.
1, 2, . . ., p. 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 Kågstr öm 39 proposed recursive block algorithms for solving the one-sided and coupled Sylvester matrix equations AX − Y B, DX − Y E C, F .Jonsson and Kågstr öm 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 AY −ZB, CY −ZD E, F .Very recently, Dehghan and Hajarian 12 constructed an iterative algorithm to solve the generalized coupled Sylvester matrix equations AXB CY D, EXF GY H M, N over generalized bisymmetric matrices.Huang et al. 13 present an iterative algorithm for the generalized coupled Sylvester matrix equations AY − ZB, CY − ZD E, F 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 AXB −CY D, EXF − GY H 1 , the generalized Sylvester matrix equation AX − Y B, CX − Y D E, F 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 .