Solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices

Abstract

In many areas of principal component analysis, biology, electricity, solid mechanics, automatics control theory and vibration theory, linear matrix equations can be encountered. The presented paper proposes first an iterative method for finding the generalized bisymmetric solution to the generalized coupled Sylvester matrix equations. Second, when the generalized coupled Sylvester matrix equations matrix are consistent, for any generalized bisymmetric initial iterative matrix pair, we can obtain the generalized bisymmetric solution within finite iterative steps in the absence of round-off errors. Furthermore, the optimal approximate generalized bisymmetric solution of the matrix equation for a given generalized bisymmetric matrix can be obtained by finding the least-norm generalized bisymmetric solution of new generalized coupled Sylvester matrix equations. Finally, numerical examples are presented to support the theoretical results of this paper.

Keywords

Coupled Sylvester matrix equations iterative method;bisymmetric solution least norm generalized solution optimal approximation solution

Introduction

Linear matrix equations play important roles in many areas of applied sciences for example in control system theory, biology, electricity, solid mechanics, automatic control theory and vibration theory. We first give some notations, which are used throughout the paper. The notations $R^{m \times n}$ , $S R^{n \times n}$ , $SO R^{n \times n}$ and $BS R^{n \times n} (U)$ represent the set of all $m \times n$ real, $n \times n$ real symmetric matrices, $n \times n$ real symmetric orthogonal matrices and $n \times n$ real generalized bisymmetric with respect to the $n \times n$ generalized reflection matrix $U$ , respectively . We use $A^{T}$ and $tr (A)$ and $R (A)$ to denote the transpose, the trace and the column space of the matrix $A$ , respectively. $vec (A)$ represents the $vec$ operator of $A$ , i.e. $vec (A) = {(a_{1}^{T}, a_{2}^{T}, \dots, a_{n}^{T})}^{T}$ for the matrix $A = (a_{1}, a_{2}, \dots, a_{n}) \in R^{m \times n}, a_{i} \in R^{m}$ , $i = 1, 2, \dots, n$ and $A \otimes B$ stands for the Kronecker product of the matrices $A$ and $B$ . Defining an inner product as $〈 A, B 〉 = tr (B^{T} A),$ then the norm of a matrix $A$ generated by this inner product is known as the Forbenius norm of $A$ and is denoted by $‖ A ‖$ (Datta, 2003; Golub and Van Loan, 1996).

For simplicity, we also introduce the following operators:

f (X, Y) = AX + BY - EYF,

g (X, Y) = MX + NY - GYH,

where $X$ and $Y$ are arbitrary matrices of appropriate dimensions with real entries.

Definition 1. (Dehghan and Hajarian, 2010a). For an arbitrary given matrix $U \in SO R^{n \times n}$ , i.e. $U = U^{T} = U^{- 1}$ , an $n \times n$ real matrix $A$ is said to be generalized bisymmetric matrix with respect to the $n \times n$ generalized reflection matrix $U$ if $A = A^{T} = UAU$ , and is denoted by $A \in BS R^{n \times n} (U)$ .

In this paper, we will consider the following four problems:

Problem 1. For given $A, E \in R^{m \times p}$ , $B \in R^{m \times q}$ , $F, H \in R^{n \times n}$ , $C \in R^{m \times n}$ $M, G \in R^{r \times p},$ $N \in R^{r \times q},$ and $D \in R^{r \times n},$ find matrix pair $[V, W]$ with $V \in R^{p \times n}$ and $W \in R^{q \times n}$ such that

{\begin{matrix} AV + BW = EVF + C, \\ MV + NW = GVH + D . \end{matrix}

(1)

Problem 2. When Problem 1 is consistent, let its solution pair set be denoted by $S_{1}$ . For a given matrix pair [ $\hat{V}, \hat{W}$ ] with $\hat{V} \in R^{p \times n}$ and $\hat{W} \in R^{q \times n}$ find matrix pair $[\tilde{V}, \tilde{W}] \in S_{1}$ such that

{‖ \tilde{V} - \hat{V} ‖}^{2} + {‖ \tilde{W} - \hat{W} ‖}^{2} = min_{V, W \in S_{1}} {{‖ V - \hat{V} ‖}^{2} + {‖ W - \hat{W} ‖}^{2}} .

(2)

In fact, Problem 2 is to find the optimal approximation solution to the given matrix pair [ $\hat{V}, \hat{W}$ ].

Problem 3. For given $A, B, E, C \in R^{m \times n}$ , $F, H \in R^{n \times n}$ and $M, N, G, D \in R^{p \times n}$ , find matrix pair $[V, W]$ with $V, W \in BS R^{n \times n},$ such that Equations (1) are satisfied.

Problem 4. When Problem 3 is consistent, let its solution pair set be denoted by $S_{2}$ . For a given matrix pair [ $\hat{V}, \hat{W}$ ] with $\hat{V}, \hat{W} \in BS R^{n \times n}$ find matrix pair $[\tilde{V}, \tilde{W}] \in S_{2}$ such that

{‖ \tilde{V} - \hat{V} ‖}^{2} + {‖ \tilde{W} - \hat{W} ‖}^{2} = min_{V, W \in S_{2}} {{‖ V - \hat{V} ‖}^{2} + {‖ W - \hat{W} ‖}^{2}} .

(3)

Notice that Problem 4 is to find the optimal approximation solution to the given matrix pair [ $\hat{V}, \hat{W}$ ].

It is well known that matrix equations are one of the topics that have many applications in various areas of computational mathematics, control and system theory. In particular, the generalized Sylvester matrix equations are closely related to many problems in descriptor linear systems such as eigenstructure assignment (Duan, 1992, 1998; Fletcher et al., 1986) and observer design (Carvalho and Datta, 2002; Castelan and da Silva, 2005; Wu et al., 2007). A large number of papers have presented several methods for solving matrix equations (Datta, 1995, 2003; Duan, 1993; Golub et al., 1979; Hou et al., 2006; Kaabi, 2007; Wu et al., 2008a, 2008b; Xing ping and Guoliang, 2007; Bartels and Stewart, 1972). One of these methods is the iterative method, which is commonly employed to solve large linear matrix equations.

An iterative method for a symmetric solution of the matrix equation $AXB = C$ is proposed by Peng et al. (2005) and the system of matrix equations $A_{1} X B_{1} = C_{1}, A_{2} X B_{2} = C_{2}$ bye Peng et al. (2006). The iterative solutions of the matrix equation $AXB = F$ and the generalized Sylvester matrix equation $AXB + CXD = F$ , by extending Jacobi and Gauss–Seidel iteration methods for $Ax = b$ , are obtained by Ding et al. (2008). Zhou and Duan (2006, 2007) presented the solution of several generalized Sylvester matrix equations, Monsalve M (2008), proposed block linear methods for large scale Sylvester matrix equations XA+BX = C. In addition, Zhou et al. (2009) proposed gradient-based iterative algorithms for solving the general coupled Sylvester matrix equations with weighted least squares solutions. Huang et al. (2008) proposed a new iterative method for the skew–symmetric solution of the matrix equation $AXB = C$ . In Li and Wu (2008), symmetric and skew–anti-symmetric solutions to the matrix equations $A_{1} X = C_{1}, X B_{2} = C_{2}$ over the real quaternion algebra H are presented. Wang et al. (2007) introduced an iterative algorithm for solving the matrix equation $AXB + C X^{T} D = E$ . Dehghan and Hajarian (2008a) proposed an iterative method for solving the pair of matrix equations $AXB = E$ and $CYD = F$ over generalized centro-symmetric matrices, finite iterative algorithms for the reflexive and anti-reflexive solutions of coupled Sylvester matrix equations (Dehghan and Hajarian, 2008b, 2009b, 2010a), and an iterative method for solving generalized coupled Sylvester matrix equations over generalized bisymmetric matrices (Dehghan and Hajarian, 2010b) . Dehghan and Hajarian (2009c, 2010c), proposed the necessary and sufficient conditions for the solvability of matrix equations

A_{1} X B_{1} = C_{1},

A_{1} X = C_{1}, X B_{2} = C_{2},

A_{1} X = C_{1}, X B_{2} = C_{2}, A_{3} X = C_{3}, X B_{4} = C_{4}

and

AXB + CYD = E

over the reflexive and anti-reflexive matrices, and obtained the general expression of the solutions for a solvable case. Also, Dehghan and Hajarian (2009a) proposed an efficient iterative method for solving the second-order Sylvester matrix equation

EV F^{2} - AVF - CV = BW

To solve (coupled) matrix equations, based on the iterative methods of matrix equations, Ding and Chen presented the hierarchical gradient iterative (HGI) algorithms for general matrix equations (Ding and Chen, 2005a; Ding et al., 2008) and hierarchical least squares iterative (HLSI) algorithms for general coupled matrix equations and generalized coupled Sylvester matrix equations (Ding and Chen, 2005a, 2005b). The HGI algorithms (Ding and Chen, 2005c; Ding et al., 2008) and HLSI algorithms (Ding and Chen, 2005d, 2006; Ding et al., 2008) for solving general (coupled) matrix equations are innovative and computationally efficient numerical algorithms, and were presented based on the hierarchical identification principle (Ding and Chen, 2005d, 2006), which considers the unknown matrix as the system parameter matrix to be identified.

This paper is organized as follows: in the next section, we propose an iterative algorithm and its properties for solving the generalized coupled Sylvester matrix equations (1). By using the proposed algorithm, we prove that for any initial matrix pair $[V_{1}, W_{1}]$ , a generalized solution can be obtained within finite iterative steps. Also the least Frobenius norm generalized solution pair can be obtained by choosing a special kind of initial matrix pair. Then, the optimal generalized solution pair of Problem 2 is derived by finding the least Frobenius norm generalized solution pair of the new generalized coupled Sylvester matrix equations. We also propose an iterative algorithm and its properties for solving the generalized coupled Sylvester matrix equations (1) but over generalized bisymmetric (symmetric) matrices. Furthermore, the optimal generalized bisymmetric (symmetric) solution pair of Problem 4 is also introduced. Finally, some numerical examples are given to illustrate the accuracy of the proposed algorithms.

We would like to comment that the matrix equation (1) is quite general and includes many matrix equations. Also, the novelty and the contribution of this work is that the four considered problems stated above are of more general forms for generalized coupled Sylvester matrix equations than most recent Sylvester matrix equations available in the literature. Such considered forms need some effort to study their solutions over generalized bisymmetric (symmetric) matrices.

An iterative algorithm for solving Problem 1

In this section, we construct an iterative algorithm for solving the generalized coupled Sylvester matrix equations (1). Then we introduce some basic properties of the iterative algorithm. Finally, we prove that the iteration will stop within finite steps.

Algorithm 1

Step 1: Input matrices $A, E \in R^{m \times p}$ , $B \in R^{m \times q}$ , $F, H \in R^{n \times n}$ , $C \in R^{m \times n}$ , $M, G \in R^{r \times p}$ , $N \in R^{r \times q}$ , $D \in R^{r \times n},$ and matrix pair $[V_{1}, W_{1}]$ with $V_{1} \in R^{p \times n}$ and $W_{1} \in R^{q \times n}$ ;

Step 2: Compute

$R_{1} = diag (C - f (V_{1}, W_{1}), D - g (V_{1}, W_{1}));$

$\begin{matrix} S_{1} = A^{T} [C - f (V_{1}, W_{1})] - E^{T} [C - f (V_{1}, W_{1})] F^{T} + M^{T} [D - g (V_{1}, W_{1})] \\ - G^{T} [D - g (V_{1}, W_{1})] H^{T}; \end{matrix}$

$T_{1} = B^{T} [C - f (V_{1}, W_{1})] + N^{T} [D - g (V_{1}, W_{1})];$

$k : = 1;$

Step 3: If $R_{k} = 0,$ then stop and $[V_{k}, W_{k}]$ is the solution; else if $R_{k} \neq 0$ but $S_{k} = 0$ and $T_{k} = 0,$ then stop and the generalized coupled Sylvester matrix equations (1) are not consistent; else $k : = k + 1;$

Step 4: Compute

$V_{k} = V_{k - 1} + \frac{{‖ R_{k - 1} ‖}^{2}}{{‖ S_{k - 1} ‖}^{2} + {‖ T_{k - 1} ‖}^{2}} S_{k - 1};$

$W_{k} = W_{k - 1} + \frac{{‖ R_{k - 1} ‖}^{2}}{{‖ S_{k - 1} ‖}^{2} + {‖ T_{k - 1} ‖}^{2}} T_{k - 1};$

$R_{k} = diag (C - f (V_{k}, W_{k}), D - g (V_{k}, W_{k}))$

$= R_{k - 1} - \frac{{‖ R_{k - 1} ‖}^{2}}{{‖ S_{k - 1} ‖}^{2} + {‖ T_{k - 1} ‖}^{2}} diag (f (S_{k - 1}, T_{k - 1}), g (S_{k - 1}, T_{k - 1}));$

$\begin{matrix} S_{k} = A^{T} [C - f (V_{k}, W_{k})] - E^{T} [C - f (V_{k}, W_{k})] F^{T} + M^{T} [D - g (V_{k}, W_{k})] \\ - G^{T} [D - g (V_{k}, W_{k})] H^{T} + \frac{{‖ R_{k} ‖}^{2}}{{‖ R_{k - 1} ‖}^{2}} S_{k - 1}; \end{matrix}$

$T_{k} = B^{T} [C - f (V_{k}, W_{k})] + N^{T} [D - g (V_{k}, W_{k})] + \frac{{‖ R_{k} ‖}^{2}}{{‖ R_{k - 1} ‖}^{2}} T_{k - 1};$

Step 5: Go to step 3.

Remark 1. It is obvious from Algorithm 1 that $V_{k}, S_{k} \in R^{p \times n}$ and $W_{k}, T_{k} \in R^{q \times n}$ where $k = 1, 2, \dots$ .

Lemma 1. For the sequences ${R_{i}}, {S_{i}}$ and ${T_{i}}$ $(i = 1, 2, \dots, s, R_{i} \neq 0)$ generated by Algorithm 1, we have

tr (R_{j}^{T} R_{i}) = 0, tr (S_{j}^{T} S_{i} + T_{j}^{T} T_{i}) = 0, for i, j = 1, 2, \dots, k, i \neq j .

(4)

The proof of the Lemma 1 is presented in the Appendix.

Lemma 2. Suppose that Problem 1 is consistent, and [ $V^{*}, W^{*}$ ] is an arbitrary solution pair of it. Then for any initial generalized matrix pair $[V_{1}, W_{1}]$ with $V_{1} \in R^{p \times n}$ and $W_{1} \in R^{q \times n}$ , we have

tr [(V^{*} - V_{i}) S_{i}^{T} + (W^{*} - W_{i}) T_{i}^{T}] = {‖ R_{i} ‖}^{2}, i = 1, 2, \dots

(5)

where the sequences ${V_{i}}, {W_{i}}, {S_{i}}, {T_{i}}$ and ${R_{i}}$ are generated by Algorithm 1.

The proof of the Lemma 2 is also presented in the Appendix.

Remark 2. Obviously, from Lemma 2, if there exists a positive integer k, such that $S_{k} = 0$ and $T_{k} = 0$ but $R_{k} \neq 0$ , we have that the matrix equations (1) are not consistent. Hence the solvability of the generalized coupled Sylvester matrix equations can be determined automatically by Algorithm 1 in the absence of round-off errors.

Theorem 1. Let Problem 1 be consistent, then for any arbitrary initial generalized matrix pair $[V_{1}, W_{1}]$ , with $V_{1} \in R^{p \times n}$ and $W_{1} \in R^{q \times n}$ , a generalized solution can be obtained within finite iterative steps in the absence of round-off errors.

Proof. Suppose that $R_{i} \neq 0$ for $i = 1, 2, \dots, mn + rn .$ From Lemma 2, we have $S_{i} \neq 0$ or $T_{i} \neq 0$ for $i = 1, 2, \dots, mn + rn$ then we can compute $R_{mn + rn + 1}$ and $[V_{mn + rn + 1}, W_{mn + rn + 1}]$ by Algorithm 1. Also from Lemma 1, we can write

tr (R_{mn + rn + 1}^{T} R_{i}) = 0 for i = 1, 2, \dots, mn + rn,

(6)

and

tr (R_{i}^{T} R_{j}) = 0 for i, j = 1, 2, \dots, mn + rn, (i \neq j) .

(7)

Therefore, the set ${R_{1}, R_{2}, \dots, R_{mn + rn}}$ is an orthogonal basis of the subspace $S = {M | M = diag (M_{1}, M_{2}), where M_{1} \in R^{m \times n} and M_{2} \in R^{r \times n}}$ of the matrix space $R^{(m + r) \times (2 n)}$ . Hence $R_{mn + rn . + 1} = 0$ and $[V_{mn + rn . + 1}, W_{mn + rn . + 1}]$ is a generalized solution of Problem 1. Therefore, the proof is completed.

Now, to obtain the least Forbenius norm solution of the generalized solution pair of Problem 1, we first present the following lemma.

Lemma 3. (Peng et al., 2006). Assume that the consistent system of linear equations $Ax = b$ has a solution $x^{*} \in R (A^{T})$ , and then $x^{*}$ is a unique least Forbenius norm solution of the system of linear equations.

Obviously, the generalized coupled Sylvester matrix equations (1) is equivalent to

[\begin{matrix} I \otimes A - F^{T} \otimes E & I \otimes B \\ I \otimes M - H^{T} \otimes G & I \otimes N \end{matrix}] [\begin{matrix} vec (V) \\ vec (W) \end{matrix}] = [\begin{matrix} vec (C) \\ vec (D) \end{matrix}] .

(8)

Now suppose $X \in R^{p \times n}$ and $Y \in R^{q \times n}$ are arbitrary matrices, then we have

[\begin{matrix} vec (A^{T} X - E^{T} X F^{T} + M^{T} Y - G^{T} Y H^{T}) \\ vec (B^{T} X + N^{T} Y) \end{matrix}]

= [\begin{matrix} I \otimes A^{T} - F \otimes E^{T} & I \otimes M^{T} - H \otimes G^{T} \\ I \otimes B^{T} & I \otimes N^{T} \end{matrix}] [\begin{matrix} vec (X) \\ vec (Y) \end{matrix}]

= {[\begin{matrix} I \otimes A - F^{T} \otimes E & I \otimes B \\ I \otimes M - H^{T} \otimes G & I \otimes N \end{matrix}]}^{T} [\begin{matrix} vec (X) \\ vec (Y) \end{matrix}]

\in R ({[\begin{matrix} I \otimes A - F^{T} \otimes E & I \otimes B \\ I \otimes M - H^{T} \otimes G & I \otimes N \end{matrix}]}^{T}) .

Obviously, if we consider

V_{1} = A^{T} X - E^{T} X F^{T} + M^{T} Y - G^{T} Y H^{T},

(9)

W_{1} = B^{T} X + N^{T} Y,

(10)

then all $V_{k}$ and $W_{k}$ obtained by Algorithm 1 satisfy

[\begin{matrix} vec (V_{k}) \\ vec (W_{k}) \end{matrix}] \in R ({[\begin{matrix} I \otimes A - F^{T} \otimes E & I \otimes B \\ I \otimes M - H^{T} \otimes G & I \otimes N \end{matrix}]}^{T}) .

Hence by Lemma 3, with the initial generalized matrices (9) and (10) where $X$ and $Y$ are arbitrary, or especially, $V_{1} = 0$ and $W_{1} = 0$ , then generalized solution pair [ $V^{*}, W^{*}$ ] obtained by Algorithm 1 is the least Frobenius norm solution pair of equations (1).

From the above conclusion, we present the following theorem.

Theorem 2. Assume that Problem 1 is consistent. If we take the initial generalized matrices (9) and (10) where $X$ and $Y$ are arbitrary, or especially, $V_{1} = 0$ and $W_{1} = 0$ , then generalized solution pair [ $V^{*}, W^{*}$ ] generated by Algorithm 1 is the least Frobenius norm generalized solution pair of Equations (1).

The solution of Problem 2

In this section, we show that the optimal approximation solution of Problem 2 for a given generalized matrix can be obtained by finding the least Frobenius norm generalized solution of new generalized coupled Sylvester matrix equations.

Suppose that Problem 1 is consistent. Obviously the solution pair set $S_{1}$ of Problem 1 is not empty, then for a given generalized matrix pair [ $\hat{V}, \hat{W}$ ] with $\hat{V} \in R^{p \times n}$ and $\hat{W} \in R^{q \times n}$ , we can get

{\begin{matrix} AV + BW = EVF + C, \\ MV + NW = GVH + D, \end{matrix} \Leftrightarrow {\begin{matrix} A (V - \hat{V}) + B (W - \hat{W}) = E (V - \hat{V}) F + C - A \hat{V} - B \hat{W} + E \hat{V} F, \\ M (V - \hat{V}) + N (W - \hat{W}) = G (V - \hat{V}) H + D - M \hat{V} - N \hat{W} + G \hat{V} H . \end{matrix}

Let $\bar{V} = V - \hat{V}, \bar{W} = W - \hat{W}, \bar{C} = C - A \hat{V} - B \hat{W} + E \hat{V} F, \bar{D} = D - M \hat{V} - N \hat{W} + G \hat{V} H,$ then the matrix nearness Problem 2 is equivalent to first finding the least Frobenius norm generalized solution pair $[{\bar{V}}^{*}, {\bar{W}}^{*}]$ of the generalized coupled Sylvester matrix equations

{\begin{matrix} A \bar{V} + B \bar{W} = E \bar{V} F + \bar{C}, \\ M \bar{V} + N \bar{W} = G \bar{V} H + \bar{D}, \end{matrix}

(11)

which can be obtained by Algorithm 1, and initial matrix pair $[{\bar{V}}_{1}, {\bar{W}}_{1}]$ with

{\bar{V}}_{1} = A^{T} X - E^{T} X F^{T} + M^{T} Y - G^{T} Y H^{T},

(12)

{\bar{W}}_{1} = B^{T} X + N^{T} Y,

(13)

where $X$ and $Y$ are arbitrary, or especially, ${\bar{V}}_{1} = 0$ and ${\bar{W}}_{1} = 0$ . Once the matrix pair $[{\bar{V}}^{*}, {\bar{W}}^{*}]$ is obtained, the solution $[\tilde{V}, \tilde{W}]$ of the nearness Problem 2 can be represented as

[\tilde{V}, \tilde{W}] = [{\bar{V}}^{*} + \hat{V}, {\bar{W}}^{*} + \hat{W}]

(14)

An iterative algorithm for solving Problem 3

In this part, we construct an iterative method for solving the generalized coupled Sylvester matrix equations (1) over generalized bisymmetric matrices. Then we introduce some basic properties of the iterative method. Finally, we prove that the iteration will stop within finite steps.

Algorithm 2

Step 1: Input matrices $A, B, E, C \in R^{m \times n}$ , $F, H \in R^{n \times n}$ , $M, N, G, D \in R^{p \times n}$ , $U \in SO R^{n \times n}, Z \in SO R^{n \times n}$ and $V_{1} \in BS R^{n \times n} (U), W_{1} \in BS R^{n \times n} (Z)$ ;

Step 2: Compute

R_{1} = diag (C - f (V_{1}, W_{1}), D - g (V_{1}, W_{1})); \begin{matrix} P_{1} = A^{T} (C - f (V_{1}, W_{1})) - E^{T} (C - f (V_{1}, W_{1})) F^{T} + M^{T} (D - g (V_{1}, W_{1})) \\ - G^{T} (D - g (V_{1}, W_{1})) H^{T}; \end{matrix} Q_{1} = B^{T} (C - f (V_{1}, W_{1})) + N^{T} (D - g (V_{1}, W_{1})); S_{1} = \frac{1}{4} (P_{1} + P_{1}^{T} + U P_{1} U + {UP}_{1}^{T} U); T_{1} = \frac{1}{4} (Q_{1} + Q_{1}^{T} + Z Q_{1} Z + {ZQ}_{1}^{T} Z); k : = 1;

Step 4: Compute

V_{k} = V_{k - 1} + \frac{{‖ R_{k - 1} ‖}^{2}}{{‖ S_{k - 1} ‖}^{2} + {‖ T_{k - 1} ‖}^{2}} S_{k - 1}; W_{k} = W_{k - 1} + \frac{{‖ R_{k - 1} ‖}^{2}}{{‖ S_{k - 1} ‖}^{2} + {‖ T_{k - 1} ‖}^{2}} T_{k - 1}; R_{k} = diag (C - f (V_{k}, W_{k}), D - g (V_{k}, W_{k})) = R_{k - 1} - \frac{{‖ R_{k - 1} ‖}^{2}}{{‖ S_{k - 1} ‖}^{2} + {‖ T_{k - 1} ‖}^{2}} diag (f (S_{k - 1}, T_{k - 1}), g (S_{k - 1}, T_{k - 1})); \begin{matrix} P_{k} = A^{T} (C - f (V_{k}, W_{k})) - E^{T} (C - f (V_{k}, W_{k})) F^{T} + M^{T} (D - g (V_{k}, W_{k})) \\ - G^{T} (D - g (V_{k}, W_{k})) H^{T}; \end{matrix} Q_{k} = B^{T} (C - f (V_{k}, W_{k})) + N^{T} (D - g (V_{k}, W_{k})); S_{k} = \frac{1}{4} (P_{k} + P_{k}^{T} + U P_{k} U + {UP}_{k}^{T} U) + \frac{{‖ R_{k} ‖}^{2}}{{‖ R_{k - 1} ‖}^{2}} S_{k - 1}; T_{k} = \frac{1}{4} (Q_{k} + Q_{k}^{T} + Z Q_{k} Z + {ZQ}_{k}^{T} Z) + \frac{{‖ R_{k} ‖}^{2}}{{‖ R_{k - 1} ‖}^{2}} T_{k - 1};

Step 5: Go to step 3.

Remark 3. Obviously, from Algorithm 2 we have $V_{k}, S_{k} \in BS R^{n \times n} (U)$ and $W_{k}, T_{k}$ $\in BS R^{n \times n} (Z),$ where $k = 1, 2, \dots$

Lemma 4. Suppose that the sequences ${R_{i}}, {S_{i}}$ and ${T_{i}}$ $(i = 1, 2, \dots, s, R_{i} \neq 0)$ generated by the Algorithm 2, then we have

tr (R_{j}^{T} R_{i}) = 0, tr (S_{j}^{T} S_{i} + T_{j}^{T} T_{i}) = 0, for i, j = 1, 2, \dots, s, i \neq j .

(15)

The proof of the Lemma 4 is presented in the Appendix.

Lemma 5. Assume that Problem 3 is consistent, and [ $V^{*}, W^{*}$ ] is an arbitrary solution pair of Problem 3. Then for any initial generalized bisymmetric matrix pair $[V_{1}, W_{1}]$ with $V_{1} \in BS R^{n \times n} (U)$ and $W_{1} \in BS R^{n \times n} (Z),$

tr [(V^{*} - V_{i}) S_{i} + (W^{*} - W_{i}) T_{i}] = {‖ R_{i} ‖}^{2}, i = 1, 2, \dots

(16)

where the sequences ${V_{i}}, {W_{i}}, {S_{i}}, {T_{i}}$ and ${R_{i}}$ , are obtained by Algorithm 2.

The proof of the Lemma 4 is also presented in the Appendix.

Remark 4. Lemma 5 implies that if there exists a positive integer k, such that $S_{k} = 0$ and $T_{k} = 0$ but $R_{k} \neq 0$ , then the generalized coupled Sylvester matrix equations (1) are inconsistent over generalized bisymmetric pair $[V, W]$ . Hence, the solvability of the matrix equations (1) over generalized bisymmetric pair $[V, W]$ can be determined automatically by Algorithm 2 in the absence of round-off errors.

Theorem 3. Suppose that Problem 3 is consistent, then for any arbitrary initial generalized bisymmetric matrix pair $[V_{1}, W_{1}]$ , with $V_{1} \in BS R^{n \times n} (U)$ and $W_{1} \in BS R^{n \times n} (Z)$ , a generalized bisymmetric solution can be obtained within a finite iterative steps by Algorithm 2 in the absence of round-off errors.

Proof. Assume that $R_{i} \neq 0$ for $i = 1, 2, \dots, mn + pn .$ From Lemma 5, we have $S_{i} \neq 0$ or $T_{i} \neq 0$ for $i = 1, 2, \dots, mn + pn$ . Therefore we can calculate $R_{mn + pn . + 1}$ and $[V_{mn + pn . + 1}, W_{mn + pn . + 1}]$ by Algorithm 2. Also from Lemma 4, we have

tr (R_{mn + pn . + 1}^{T} R_{i}) = 0 for i = 1, 2, \dots, mn + pn

(17)

and

tr (R_{i}^{T} R_{j}) = 0 for i, j = 1, 2, \dots, mn + pn, (i \neq j) .

(18)

Hence the set ${R_{1}, R_{2}, \dots, R_{mn + pn}}$ is an orthogonal basis of the subspace $S = {M | M = diag (M_{1}, M_{2}), where M_{1} \in R^{m \times n} and M_{2} \in R^{p \times n}}$ of the matrix space $R^{(m + p) \times (2 n)}$ , which implies that $R_{mn + pn . + 1} = 0$ , i.e. $[V_{mn + pn . + 1}, W_{mn + pn . + 1}]$ is a generalized bisymmetric solution of Problem 3. Therefore, the proof is completed.

Lemma 6. A necessary and sufficient condition for the generalized coupled Sylvester matrix equations (1) to have a generalized bisymmetric solution is that the following matrix equations

{\begin{matrix} AV + BW = EVF + C, \\ V A^{T} + W B^{T} = F^{T} V E^{T} + C^{T}, \\ AUVU + BZWZ = EUVUF + C, \\ UVU A^{T} + ZWZ B^{T} = F^{T} UVU E^{T} + C^{T}, \\ MV + NW = GVH + D, \\ V M^{T} + W N^{T} = H^{T} V G^{T} + D^{T}, \\ MUVU + NZWZ = GUVUH + D, \\ UVU M^{T} + ZWZ N^{T} = H^{T} UVU G^{T} + D^{T}, \end{matrix}

(19)

are consistent.

Proof. First, we assume that the generalized coupled Sylvester matrix equations (1) have a generalized bisymmetric solution pair [ $V^{*}, W^{*}$ ].

By ${V^{*}}^{T} = V^{*} = U V^{*} U$ , ${W^{*}}^{T} = W^{*} = Z W^{*} Z$ and

{\begin{matrix} A V^{*} + B W^{*} = E V^{*} F + C, \\ M V^{*} + N W^{*} = G V^{*} H + D, \end{matrix}

we can obtain

\begin{matrix} AU V^{*} U + BZ W^{*} Z = A V^{*} + B W^{*} \\ = E V^{*} F + C \\ = EU V^{*} UF + C, \end{matrix}

(20)

\begin{matrix} U V^{*} U A^{T} + Z W^{*} Z B^{T} = V^{*} A^{T} + W^{*} B^{T} \\ = {V^{*}}^{T} A^{T} + {W^{*}}^{T} B^{T} \\ = {(A V^{*} + B W^{*})}^{T} \\ = {(E V^{*} F + C)}^{T} \\ = F^{T} U V^{*} U E^{T} + C^{T}, \end{matrix}

(21)

\begin{matrix} MU V^{*} U + NZ W^{*} Z = M V^{*} + N W^{*} \\ = G V^{*} H + D \\ = GU V^{*} UH + D, \end{matrix}

(22)

and

\begin{matrix} U V^{*} U M^{T} + Z W^{*} Z N^{T} = V^{*} M^{T} + W^{*} N^{T} \\ = {V^{*}}^{T} M^{T} + {W^{*}}^{T} N^{T} \\ = {(M V^{*} + N W^{*})}^{T} \\ = {(G V^{*} H + D)}^{T} \\ = H^{T} U V^{*} U G^{T} + D^{T} . \end{matrix}

(23)

That is, the generalized bisymmetric solution pair [ $V^{*}, W^{*}$ ] is a solution pair of the matrix equations (19).

Conversely, assume that the matrix equations (19) are consistent. Let $[V, W]$ be a solution pair of the matrix equations (19).

Let $V_{0} = \frac{V + V^{T} + UVU + U V^{T} U}{4}, and W_{0} = \frac{W + W^{T} + ZWZ + Z W^{T} Z}{4}$ .

Hence $V_{0} \in BS R^{n \times n} (U),$ and $W_{0} \in BS R^{n \times n} (Z),$

\begin{matrix} A V_{0} + B W_{0} - E V_{0} F = A \frac{V + V^{T} + UVU + U V^{T} U}{4} + B \frac{W + W^{T} + ZWZ + Z W^{T} Z}{4} \\ - E \frac{V + V^{T} + UVU + U V^{T} U}{2} F \end{matrix} \begin{matrix} = \frac{AV + A V^{T} + AUVU + AU V^{T} U + BW + B W^{T} + BZWZ + BZ W^{T} Z}{4} \\ - \frac{EVF + E V^{T} F + EUVUF + EU V^{T} UF}{4} \end{matrix} = \frac{AV + BW - EVF + {(V A^{T})}^{T} + {(W B^{T})}^{T} - {(F^{T} V E^{T})}^{T}}{2} = \frac{C + {(C^{T})}^{T}}{2} = C .

and similarly

\begin{matrix} M V_{0} + N W_{0} - G V_{0} H = M \frac{V + V^{T} + UVU + U V^{T} U}{4} + N \frac{W + W^{T} + ZWZ + Z W^{T} Z}{4} \\ - G \frac{V + V^{T} + UVU + U V^{T} U}{4} H = \frac{D + {(D^{T})}^{T}}{2} = D . \end{matrix}

Hence, $[V_{0}, W_{0}]$ is a generalized bisymmetric solution pair of the generalized coupled Sylvester matrix equations (1). Therefore, the proof is completed.

To obtain least Frobenius norm solution of the generalized bisymmetric solution pair of Problem 3, obviously, from Lemma 6, the solvability of the matrix equations (19) is equivalent to the solvability of Problem 3. In addition, the system of matrix equations (19) is equivalent to

[\begin{matrix} I \otimes A - F^{T} \otimes E & I \otimes B \\ A \otimes I - E \otimes F^{T} & B \otimes I \\ U \otimes AU - F^{T} U \otimes EU & Z \otimes BZ \\ AU \otimes U - EU \otimes F^{T} U & BZ \otimes Z \\ I \otimes M - H^{T} \otimes G & I \otimes N \\ M \otimes I - G \otimes H^{T} & N \otimes I \\ U \otimes MU - H^{T} U \otimes GU & Z \otimes NZ \\ MU \otimes U - GU \otimes H^{T} U & NZ \otimes Z \end{matrix}] [\begin{matrix} vec (V) \\ vec (W) \end{matrix}] = [\begin{matrix} vec (C) \\ vec (C^{T}) \\ vec (C) \\ vec (C^{T}) \\ vec (D) \\ vec (D^{T}) \\ vec (D) \\ vec (D^{T}) \end{matrix}] .

(24)

Now, suppose that $X \in R^{m \times n}$ and $Y \in R^{p \times n}$ are arbitrary matrices, then we having

[\begin{matrix} vec (\begin{matrix} A^{T} X - E^{T} X F^{T} + X^{T} A - F X^{T} E + U A^{T} XU - U E^{T} X F^{T} U + U X^{T} AU - UF X^{T} EU \\ + M^{T} Y - G^{T} Y H^{T} + Y^{T} M - H Y^{T} G + U M^{T} YU - U G^{T} Y H^{T} U + U Y^{T} MU - UH Y^{T} GU \end{matrix}) \\ vec (B^{T} X + N^{T} Y + Z B^{T} XZ + Z N^{T} YZ + X^{T} B + Y^{T} N + Z X^{T} BZ + Z Y^{T} NZ) \end{matrix}]

= {[\begin{matrix} I \otimes A - F^{T} \otimes E & I \otimes B \\ A \otimes I - E \otimes F^{T} & B \otimes I \\ U \otimes AU - F^{T} U \otimes EU & Z \otimes BZ \\ AU \otimes U - EU \otimes F^{T} U & BZ \otimes Z \\ I \otimes M - H^{T} \otimes G & I \otimes N \\ M \otimes I - G \otimes H^{T} & N \otimes I \\ U \otimes MU - H^{T} U \otimes GU & Z \otimes NZ \\ MU \otimes U - GU \otimes H^{T} U & NZ \otimes Z \end{matrix}]}^{T} [\begin{matrix} vec (X) \\ vec (X^{T}) \\ vec (X) \\ vec (X^{T}) \\ vec (Y) \\ vec (Y^{T}) \\ vec (Y) \\ vec (Y^{T}) \end{matrix}]

\in R ({[\begin{matrix} I \otimes A - F^{T} \otimes E & I \otimes B \\ A \otimes I - E \otimes F^{T} & B \otimes I \\ U \otimes AU - F^{T} U \otimes EU & Z \otimes BZ \\ AU \otimes U - EU \otimes F^{T} U & BZ \otimes Z \\ I \otimes M - H^{T} \otimes G & I \otimes N \\ M \otimes I - G \otimes H^{T} & N \otimes I \\ U \otimes MU - H^{T} U \otimes GU & Z \otimes NZ \\ MU \otimes U - GU \otimes H^{T} U & NZ \otimes Z \end{matrix}]}^{T}) .

It is obvious that if we consider

\begin{matrix} V_{1} = A^{T} X - E^{T} X F^{T} + X^{T} A - F X^{T} E + U A^{T} XU - U E^{T} X F^{T} U + U X^{T} AU \\ - UF X^{T} EU + M^{T} Y - G^{T} Y H^{T} + Y^{T} M - H Y^{T} G + U M^{T} YU - U G^{T} Y H^{T} U \\ + U Y^{T} MU - UH Y^{T} GU, \end{matrix}

(25)

W_{1} = B^{T} X + N^{T} Y + Z B^{T} XZ + Z N^{T} YZ + X^{T} B + Y^{T} N + Z X^{T} BZ + Z Y^{T} NZ,

(26)

then all $V_{k}$ and $W_{k}$ generated by Algorithm 2 satisfy

[\begin{matrix} vec (V_{k}) \\ vec (W_{k}) \end{matrix}] \in R ({[\begin{matrix} I \otimes A - F^{T} \otimes E & I \otimes B \\ A \otimes I - E \otimes F^{T} & B \otimes I \\ U \otimes AU - F^{T} U \otimes EU & Z \otimes BZ \\ AU \otimes U - EU \otimes F^{T} U & BZ \otimes Z \\ I \otimes M - H^{T} \otimes G & I \otimes N \\ M \otimes I - G \otimes H^{T} & N \otimes I \\ U \otimes MU - H^{T} U \otimes GU & Z \otimes NZ \\ MU \otimes U - GU \otimes H^{T} U & NZ \otimes Z \end{matrix}]}^{T}) .

Hence, considering Lemma 3, with the initial generalized bisymmetric matrices (25) and (26) where $X$ and $Y$ are arbitrary, or especially, $V_{1} = 0$ and $W_{1} = 0$ , the generalized bisymmetric solution pair [ $V^{*}, W^{*}$ ] generated by Algorithm 2 is the least Frobenius norm bisymmetric solution pair.

From the above conclusion, we present the following theorem.

Theorem 4. Suppose that Problem 3 is consistent, if we take the initial generalized bisymmetric matrices (25) and (26), where $X$ and $Y$ are arbitrary, or especially, $V_{1} = 0$ and $W_{1} = 0$ , then generalized bisymmetric solution pair [ $V^{*}, W^{*}$ ] generated by Algorithm 2 is the least Frobenius norm generalized bisymmetric solution pair.

Remark 5. We know that the identity matrix I is a symmetric orthogonal matrix, because using Algorithm 2 we can obtain a symmetric solution pair of the generalized coupled Sylvester matrix equations (1) by putting $U = Z = I$ .

In the symmetric case, we can obtain the following algorithm.

Algorithm 3

Step 1: Input matrices $A, B, E, C \in R^{m \times n}$ , $F, H \in R^{n \times n}$ , $M, N, G, D \in R^{p \times n}$ , and

V_{1}, W_{1} \in S R^{n \times n};

Step 2: Compute

R_{1} = diag (C - f (V_{1}, W_{1}), D - g (V_{1}, W_{1})); \begin{matrix} P_{1} = A^{T} (C - f (V_{1}, W_{1})) - E^{T} (C - f (V_{1}, W_{1})) F^{T} + M^{T} (D - g (V_{1}, W_{1})) \\ - G^{T} (D - g (V_{1}, W_{1})) H^{T}; \end{matrix} Q_{1} = B^{T} (C - f (V_{1}, W_{1})) + N^{T} (D - g (V_{1}, W_{1})); S_{1} = \frac{1}{2} (P_{1} + P_{1}^{T}); T_{1} = \frac{1}{2} (Q_{1} + Q_{1}^{T}); k : = 1;

Step 3: If $R_{k} = 0,$ then stop and $[V_{k}, W_{k}]$ is the solution; else if $R_{k} \neq 0,$ but $S_{k} = 0$ and $T_{k} = 0,$ then stop and the generalized coupled Sylvester matrix equations (1) are not consistent over symmetric matrices; else

$k : = k + 1$ ;

Step 4: Compute

V_{k} = V_{k - 1} + \frac{{‖ R_{k - 1} ‖}^{2}}{{‖ S_{k - 1} ‖}^{2} + {‖ T_{k - 1} ‖}^{2}} S_{k - 1}; W_{k} = W_{k - 1} + \frac{{‖ R_{k - 1} ‖}^{2}}{{‖ S_{k - 1} ‖}^{2} + {‖ T_{k - 1} ‖}^{2}} T_{k - 1}; R_{k} = diag (C - f (V_{k}, W_{k}), D - g (V_{k}, W_{k})) = R_{k - 1} - \frac{{‖ R_{k - 1} ‖}^{2}}{{‖ S_{k - 1} ‖}^{2} + {‖ T_{k - 1} ‖}^{2}} diag (f (S_{k - 1}, T_{k - 1}), g (S_{k - 1}, T_{k - 1})); \begin{matrix} P_{k} = A^{T} (C - f (V_{k}, W_{k})) - E^{T} (C - f (V_{k}, W_{k})) F^{T} + M^{T} (D - g (V_{k}, W_{k})) \\ - G^{T} (D - g (V_{k}, W_{k})) H^{T}; \end{matrix} Q_{k} = B^{T} (C - f (V_{k}, W_{k})) + N^{T} (D - g (V_{k}, W_{k})); S_{k} = \frac{1}{2} (P_{k} + P_{k}^{T}) + \frac{{‖ R_{k} ‖}^{2}}{{‖ R_{k - 1} ‖}^{2}} S_{k - 1}; T_{k} = \frac{1}{2} (Q_{k} + Q_{k}^{T}) + \frac{{‖ R_{k} ‖}^{2}}{{‖ R_{k - 1} ‖}^{2}} T_{k - 1};

Step 5: Go to step 3.

We can also obtain the following notes in the symmetric case:

A necessary and sufficient condition that the generalized coupled Sylvester matrix equations (1) have a generalized symmetric solution is that the following matrix equations

{\begin{matrix} AV + BW = EVF + C, \\ V A^{T} + W B^{T} = F^{T} V E^{T} + C^{T}, \\ MV + NW = GVH + D, \\ V M^{T} + W N^{T} = H^{T} V G^{T} + D^{T}, \end{matrix}

(27)

are consistent.

To obtain the least Frobenius norm solution of the generalized symmetric solution pair of Problem 3, it is obvious that the system matrix equations (27) are equivalent to

[\begin{matrix} I \otimes A - F^{T} \otimes E & I \otimes B \\ A \otimes I - E \otimes F^{T} & B \otimes I \\ I \otimes M - H^{T} \otimes G & I \otimes N \\ M \otimes I - G \otimes H^{T} & N \otimes I \end{matrix}] [\begin{matrix} vec (V) \\ vec (W) \end{matrix}] = [\begin{matrix} vec (C) \\ vec (C^{T}) \\ vec (D) \\ vec (D^{T}) \end{matrix}] .

(28)

Now, if we consider

V_{1} = A^{T} X - E^{T} X F^{T} + M^{T} Y - G^{T} Y H^{T} + X^{T} A - F X^{T} E + Y^{T} M - H Y^{T} G,

(29)

W_{1} = B^{T} X + N^{T} Y + X^{T} B + Y^{T} N,

(30)

where $X$ and $Y$ are arbitrary, then all $V_{k}$ and $W_{k}$ generated by Algorithm 3 satisfy

[\begin{matrix} vec (V_{k}) \\ vec (W_{k}) \end{matrix}] \in R ({[\begin{matrix} I \otimes A - F^{T} \otimes E & I \otimes B \\ A \otimes I - E \otimes F^{T} & B \otimes I \\ I \otimes M - H^{T} \otimes G & I \otimes N \\ M \otimes I - G \otimes H^{T} & N \otimes I \end{matrix}]}^{T}) .

Hence by Lemma 3, with the initial generalized symmetric matrices (29) and (30) where $X$ and $Y$ are arbitrary, or especially, $V_{1} = 0$ and $W_{1} = 0$ , the generalized symmetric solution pair [ $V^{*}, W^{*}$ ] obtained by Algorithm 3 is the least Frobenius norm symmetric solution pair.

The solution of Problem 4

In this section, we show that the optimal approximation solution of Problem 4 for a given generalized bisymmetric matrix can be derived by finding the least Frobenius norm generalized bisymmetric solution of new generalized coupled Sylvester matrix equations.

Let Problem 3 be consistent. Thus the solution pair set $S_{2}$ of Problem 3 is non-empty; then for a given generalized bisymmetric matrix pair $[\hat{V}, \hat{W}]$ with $\hat{V} \in BS R^{n \times n} (U),$ and $\hat{W} \in BS R^{n \times n} (Z),$ we have

{\begin{matrix} AV + BW = EVF + C, \\ MV + NW = GVH + D, \end{matrix} \Leftrightarrow {\begin{matrix} A (V - \hat{V}) + B (W - \hat{W}) = E (V - \hat{V}) F + C - A \hat{V} - B \hat{W} + E \hat{V} F, \\ M (V - \hat{V}) + N (W - \hat{W}) = G (V - \hat{V}) H + D - M \hat{V} - N \hat{W} + G \hat{V} H . \end{matrix}

Set $\bar{V} = V - \hat{V}, \bar{W} = W - \hat{W}, \bar{C} = C - A \hat{V} - B \hat{W} + E \hat{V} F, \bar{D} = D - M \hat{V} - N \hat{W} + G \hat{V} H,$ then the matrix nearness Problem 4 is equivalent to first finding the least Frobenius norm generalized bisymmetric solution pair $[{\bar{V}}^{*}, {\bar{W}}^{*}]$ of the generalized coupled Sylvester matrix equations

{\begin{matrix} A \bar{V} + B \bar{W} = E \bar{V} F + \bar{C}, \\ M \bar{V} + N \bar{W} = G \bar{V} H + \bar{D}, \end{matrix}

(31)

by using Algorithm 2 and initial matrix pair $[{\bar{V}}_{1}, {\bar{W}}_{1}]$ with

\begin{matrix} {\bar{V}}_{1} = A^{T} X - E^{T} X F^{T} + X^{T} A - F X^{T} E + U A^{T} XU - U E^{T} X F^{T} U + U X^{T} AU \\ - UF X^{T} EU + M^{T} Y - G^{T} Y H^{T} + Y^{T} M - H Y^{T} G + U M^{T} YU - U G^{T} Y H^{T} U \\ + U Y^{T} MU - UH Y^{T} GU, \end{matrix}

(32)

{\bar{W}}_{1} = B^{T} X + N^{T} Y + Z B^{T} XZ + Z N^{T} YZ + X^{T} B + Y^{T} N + Z X^{T} BZ + Z Y^{T} NZ,

(33)

where $X$ and $Y$ are arbitrary, or especially, ${\bar{V}}_{1} = 0$ and ${\bar{W}}_{1} = 0$ . Then the bisymmetric solution pair $[\tilde{V}, \tilde{W}]$ of the nearness Problem 4 can be expressed from the following relation

[\tilde{V}, \tilde{W}] = [{\bar{V}}^{*} + \hat{V}, {\bar{W}}^{*} + \hat{W}] .

(34)

Remark 6. In the symmetric case for a given generalized symmetric matrix pair $[\hat{V}, \hat{W}]$ with $\hat{V} \in S R^{n \times n},$ and $\hat{W} \in S R^{n \times n},$ we can obtain the optimal approximation solution by finding the least Frobenius norm generalized symmetric pair $[{\bar{V}}^{*}, {\bar{W}}^{*}]$ of the new generalized coupled Sylvester matrix equations.

{\begin{matrix} A \bar{V} + B \bar{W} = E \bar{V} F + \bar{C}, \\ M \bar{V} + N \bar{W} = G \bar{V} H + \bar{D} . \end{matrix}

(35)

where

\bar{V} = V - \hat{V}, \bar{W} = W - \hat{W}, \bar{C} = C - A \hat{V} - B \hat{W} + E \hat{V} F, \bar{D} = D - M \hat{V} - N \hat{W} + G \hat{V} H,

by applying Algorithm 3 and letting the initial pair $[{\bar{V}}_{1}, {\bar{W}}_{1}]$ with

{\bar{V}}_{1} = A^{T} X - E^{T} X F^{T} + M^{T} Y - G^{T} Y H^{T} + X^{T} A - F X^{T} E + Y^{T} M - H Y^{T} G,

(36)

{\bar{W}}_{1} = B^{T} X + N^{T} Y + X^{T} B + Y^{T} N,

(37)

where $X$ and $Y$ are arbitrary, or especially, ${\bar{V}}_{1} = 0$ and ${\bar{W}}_{1} = 0$ . The symmetric solution $[\tilde{V}, \tilde{W}]$ of the nearness Problem of this case can be computed as follows

[\tilde{V}, \tilde{W}] = [{\bar{V}}^{*} + \hat{V}, {\bar{W}}^{*} + \hat{W}] .

(38)

Numerical examples

In this section, numerical examples are given to illustrate the accuracy of the proposed algorithms.

Example 1

Consider the generalized coupled Sylvester matrix equations (1) where the matrices $A, B, E, F, M, N, G, H, C$ and $D$ are given as follows

\begin{matrix} A = [\begin{matrix} 1102 \\ 00 - 12 \\ 00 - 11 \\ 20 31 \end{matrix}], B = [\begin{matrix} 001 \\ 102 \\ 012 \\ 024 \end{matrix}], E = [\begin{matrix} 1002 \\ 0103 \\ 1204 \\ 0000 \end{matrix}], F = [\begin{matrix} 012 \\ 121 \\ 023 \end{matrix}] \\ M = [\begin{matrix} 1 02 1 \\ 0 - 21 - 1 \\ 0 12 1 \\ 0 12 3 \end{matrix}], N = [\begin{matrix} 021 \\ 101 \\ 120 \\ 023 \end{matrix}], G = [\begin{matrix} 0101 \\ 0013 \\ 0100 \\ 2310 \end{matrix}], \\ H = [\begin{matrix} 110 \\ 001 \\ 032 \end{matrix}], C = [\begin{matrix} 3 - 2 - 7 \\ 6 5 - 1 \\ 9 15 9 \\ 1521 9 \end{matrix}], D = [\begin{matrix} 8 1812 \\ - 6 - 13 \\ 10 1711 \\ 23 4526 \end{matrix}] \end{matrix}

It can be verified that the matrix equations (1) are consistent and their solutions are as follows:

V = [\begin{matrix} - 2 - 1 0 \\ - 0 - 2 - 3 \\ 1 2 1 \\ 2 1 0 \end{matrix}], W = [\begin{matrix} 0 1 2 \\ 3 2 1 \\ 2 3 1 \end{matrix}]

To find the matrix pair $[V, W]$ with $V \in R^{p \times n}$ and $W \in R^{q \times n}$ such that Equations (1) are satisfied, we let the initial matrices $V_{1} = 0$ and $W_{1} = 0$ , then applying Algorithm 1, we obtain the solutions as follows

V_{34} = [\begin{matrix} - 2.0000 - 1.0000 0.0000 \\ - 0.0000 - 2.0000 - 3.0000 \\ 1.0000 2.0000 1.0000 \\ 2.0000 1.0000 - 0.0000 \end{matrix}], W_{34} = [\begin{matrix} 0.00001.00002.0000 \\ 3.00002.00001.0000 \\ 2.00003.00001.0000 \end{matrix}]

with the corresponding residual

‖ R_{34} ‖ = ‖ diag (C - f (V_{34}, W_{34}), D - g (V_{34}, W_{34})) ‖ = 7.2802 \times 1 0^{- 10} .

The obtained results are presented in Table 1 and Figure 1, where $δ_{k} = \frac{‖ [V_{k}, W_{k}] - [V, W] ‖}{‖ [V, W] ‖}$ (relative error) and $r_{k} = ‖ R_{k} ‖$ (residual).

Table 1.

The residual and the relative error for Example 1 with $V_{1} = W_{1} = 0$ .

Iteration number k	Residual r_k	Relative error δ_k
31	9.8244×10⁻⁶	2.1707×10⁻⁷
32	2.8682×10⁻⁸	7.6001×10⁻¹⁰
33	1.2558×10⁻⁸	1.5673×10⁻¹⁰
34	7.2802×10⁻¹⁰	2.3688×10⁻¹¹

Figure 1.

The residual and the relative error for Example 1 with $V_{1} = W_{1} = 0$ .

Now let

\hat{V} = [\begin{matrix} - 1 - 1 0 \\ 0 - 1 - 1 \\ 1 1 1 \\ 1 1 0 \end{matrix}], \hat{W} = [\begin{matrix} 011 \\ 111 \\ 111 \end{matrix}]

By applying Algorithm 1 for the generalized coupled Sylvester matrix equations (11), and letting the initial pair $[{\bar{V}}_{1}, {\bar{W}}_{1}] = 0$ , we can obtain the least Frobenius norm generalized solution pair $[{\bar{V}}^{*}, {\bar{W}}^{*}]$ of the generalized coupled Sylvester matrix equations (11) as follows

\begin{matrix} {\bar{V}}^{*} = {\bar{V}}_{34} = [\begin{matrix} - 1.0000 - 0.0000 - 0.0000 \\ 0.0000 - 1.0000 - 2.0000 \\ 0.0000 1.0000 - 0.0000 \\ 1.0000 0.0000 - 0.0000 \end{matrix}], \\ {\bar{W}}^{*} = {\bar{W}}_{34} = [\begin{matrix} - 0.0000 - 0.00001.0000 \\ 2.0000 1.00000.0000 \\ 1.0000 2.00000.0000 \end{matrix}] \end{matrix}

with the corresponding residual

‖ R_{34} ‖ = ‖ diag (\bar{C} - f ({\bar{V}}_{34}, {\bar{W}}_{34}), \bar{D} - g ({\bar{V}}_{34}, {\bar{W}}_{34})) ‖ = 4.5607 \times 1 0^{- 10} .

The values of the residual $r_{k}$ and the relative error $δ_{k}$ are shown in Table 2 and Figure 2.

Table 2.

The residual and the relative error for Example 1 with ${\bar{V}}_{1} = {\bar{W}}_{1} = 0$ .

Iteration number k	Residual r_k	Relative error δ_k
31	4.8545×10⁻⁶	1.9376×10⁻⁷
32	2.1707×10⁻⁸	1.0451×10⁻⁹
33	4.1717×10⁻⁹	9.4555×10⁻¹¹
34	4.5607×10⁻¹⁰	2.4101×10⁻¹¹
35	7.3451×10⁻¹¹	4.3939×10⁻¹²

Figure 2.

The residual and the relative error for Example 1 with ${\bar{V}}_{1} = {\bar{W}}_{1} = 0$ .

Once the matrix pair $[{\bar{V}}^{*}, {\bar{W}}^{*}]$ is obtained, the solution $[\tilde{V}, \tilde{W}]$ of the nearness Problem 2 can be computed from the following relations

\begin{matrix} \tilde{V} = {\bar{V}}^{*} + \hat{V} = [\begin{matrix} - 2.0000 - 1.0000 - 0.0000 \\ 0.0000 - 2.0000 - 3.0000 \\ 1.0000 2.0000 1.0000 \\ 2.0000 1.0000 - 0.0000 \end{matrix}], \\ \tilde{W} = {\bar{W}}^{*} + \hat{W} = [\begin{matrix} - 0.00001.00002.0000 \\ 3.00002.00001.0000 \\ 2.00003.00001.0000 \end{matrix}] . \end{matrix}

with the corresponding residual

‖ R_{34} ‖ = ‖ diag (C - f (\tilde{V}, \tilde{W}), D - g (\tilde{V}, \tilde{W})) ‖ = 4.5607 \times 1 0^{- 10} .

Example 2. For a symmetric case

Consider the generalized coupled Sylvester matrix equations (1) where

\begin{matrix} A = [\begin{matrix} 1023 \\ 3221 \\ 0724 \\ 1320 \end{matrix}], B = [\begin{matrix} 203 1 \\ 043 - 2 \\ 201 3 \\ 023 1 \end{matrix}], E = [\begin{matrix} 1 - 2 0 - 1 \\ 3 1 0 3 \\ 2 1 20 \\ 1 4 - 10 \end{matrix}], \\ F = [\begin{matrix} 0 - 21 - 1 \\ - 3 02 - 2 \\ 1 1 0 1 \\ - 1 - 12 0 \end{matrix}], \\ M = [\begin{matrix} 4034 \\ - 2034 \\ 0250 \\ 5410 \end{matrix}], N = [\begin{matrix} 0 - 153 \\ 0 444 \\ 0 321 \\ 1 024 \end{matrix}], G = [\begin{matrix} 1210 \\ 0231 \\ 1024 \\ - 2021 \end{matrix}], \\ H = [\begin{matrix} - 2 0 - 3 1 \\ 0 - 3 2 - 3 \\ 1 1 1 1 \\ 1 1 2 0 \end{matrix}], C = [\begin{matrix} 8 11 2617 \\ 5740 - 4548 \\ 3027 - 1050 \\ 1933 - 1517 \end{matrix}], D = [\begin{matrix} 3127 412 \\ 1417 523 \\ 627 - 2045 \\ 2215 - 1032 \end{matrix}] \end{matrix}

It can be verified that the matrix equations (1) are consistent over symmetric matrices and their symmetric solutions are as follows:

V = [\begin{matrix} 2 1 0 1 \\ 1 - 1 0 3 \\ 0 0 1 0 \\ 1 3 0 3 \end{matrix}] and W = [\begin{matrix} 1 0 2 3 \\ 0 - 1 3 2 \\ 2 3 1 0 \\ 3 2 0 2 \end{matrix}] .

To find the matrix pair $[V, W]$ with $V, W \in S R^{n \times n}$ such that Equations (1) are satisfied, we choose the initial matrices $V_{1} = 0$ and $W_{1} = 0$ . By applying Algorithm 3, we obtain the solutions as follows

\begin{matrix} V_{31} = [\begin{matrix} 2.0000 1.0000 - 0.0000 1.0000 \\ 1.0000 - 1.0000 - 0.0000 3.0000 \\ - 0.0000 - 0.0000 1.0000 - 0.0000 \\ 1.0000 3.0000 - 0.0000 3.0000 \end{matrix}], \\ W_{31} = [\begin{matrix} 1.0000 - 0.00002.00003.0000 \\ - 0.0000 - 1.00003.00002.0000 \\ 2.0000 3.00001.00000.0000 \\ 3.0000 2.00000.00002.0000 \end{matrix}] \end{matrix}

with the corresponding residual

‖ R_{31} ‖ = ‖ diag (C - f (V_{31}, W_{31}), D - g (V_{31}, W_{31})) ‖ = 2.7749 \times 1 0^{- 11}

The obtained results are presented in Table 3 and Figure 3.

Table 3.

The residual and the relative error for Example 2 with $V_{1} = W_{1} = 0$ .

Iteration number k	Residual r_k	Relative error δ_k
27	5.0517×10⁻⁷	3.7627×10⁻⁹
28	1.5391×10⁻⁸	1.3828×10⁻¹⁰
29	5.2092×10⁻⁹	2.4146×10⁻¹¹
30	1.4803×10⁻⁹	1.2794×10⁻¹¹
31	2.7749×10⁻¹¹	2.9684×10⁻¹³

Figure 3.

The residual and the relative error for Example 2 with $V_{1} = W_{1} = 0$ .

Now let

\hat{V} = [\begin{matrix} 2 & 1 & 0 & 1 \\ 1 & - 1 & 0 & 2 \\ 0 & 0 & 1 & 0 \\ 1 & 2 & 0 & 2 \end{matrix}], \hat{W} = [\begin{matrix} 1 & 0 & 2 & 2 \\ 0 & - 1 & 2 & 2 \\ 2 & 2 & 1 & 0 \\ 2 & 2 & 0 & 2 \end{matrix}] .

By Algorithm 3 with the initial matrix pair $[{\bar{V}}_{1}, {\bar{W}}_{1}] = 0$ , we can obtain the least Frobenius norm generalized solution pair $[{\bar{V}}^{*}, {\bar{W}}^{*}]$ of the generalized coupled Sylvester matrix equations (35) as follows

\begin{matrix} {\bar{V}}^{*} = {\bar{V}}_{31} = [\begin{matrix} 0.00000.0000 0.0000 - 0.0000 \\ 0.00000.0000 0.00001.0000 \\ 0.00000.0000 - 0.00000.0000 \\ - 0.00001.0000 0.00001.0000 \end{matrix}], \\ {\bar{W}}^{*} = {\bar{W}}_{31} = [\begin{matrix} 0.00000.0000 0.0000 1.0000 \\ 0.0000 - 0.0000 1.0000 0.0000 \\ 0.0000 1.0000 - 0.0000 - 0.0000 \\ 1.0000 0.0000 - 0.0000 - 0.0000 \end{matrix}] \end{matrix}

with the corresponding residual

‖ R_{31} ‖ = ‖ diag (\bar{C} - f ({\bar{V}}_{31}, {\bar{W}}_{31}), \bar{D} - g ({\bar{V}}_{31}, {\bar{W}}_{31})) ‖ = 3.6093 \times 1 0^{- 12}

The values of the relative error $δ_{k}$ and the residual $r_{k}$ are shown in Table 4 and Figure 4.

Table 4.

The residual and the relative error for Example 2 with ${\bar{V}}_{1} = {\bar{W}}_{1} = 0$ .

Iteration number k	Residual r_k	Relative error δ_k
27	1.1499×10⁻⁸	2.8680×10⁻¹⁰
28	8.2909×10⁻⁹	1.9334×10⁻¹⁰
29	5.7582×10⁻¹⁰	1.6257×10⁻¹¹
30	4.2452×10⁻¹¹	1.4887×10⁻¹²
31	3.6093×10⁻¹²	5.8673×10⁻¹⁴

Figure 4.

The residual and the relative error for Example 2 with ${\bar{V}}_{1} = {\bar{W}}_{1} = 0$ .

The symmetric solution $[\tilde{V}, \tilde{W}]$ of the nearness problem can be calculated from the following relations.

\begin{matrix} \tilde{V} = {\bar{V}}^{*} + \hat{V} = [\begin{matrix} 2.0000 1.00000.00001.0000 \\ 1.0000 - 1.00000.00003.0000 \\ 0.0000 0.00001.00000.0000 \\ 1.0000 3.00000.00003.0000 \end{matrix}] \\ \tilde{W} = {\bar{W}}^{*} + \hat{W} = [\begin{matrix} 1.0000 0.0000 2.0000 3.0000 \\ 0.0000 - 1.0000 3.0000 2.0000 \\ 2.0000 3.0000 1.0000 - 0.0000 \\ 3.0000 2.0000 - 0.0000 2.0000 \end{matrix}] \end{matrix}

with the corresponding residual

‖ R_{31} ‖ = ‖ diag (C - f (\tilde{V}, \tilde{W}), D - g (\tilde{V}, \tilde{W})) ‖ = 3.6027 \times 1 0^{- 12}

Example 3. For a bisymmetric case

Consider the generalized coupled Sylvester matrix equations (1) with the following matrices:

\begin{matrix} A = [\begin{matrix} 1 - 263 \\ 2435 \\ 0241 \\ - 6240 \end{matrix}], B = [\begin{matrix} 3523 \\ 0231 \\ 2034 \\ 3242 \end{matrix}], E = [\begin{matrix} 3 21 - 4 \\ 0 - 42 4 \\ 1 01 - 5 \\ 2 - 45 2 \end{matrix}], \\ F = [\begin{matrix} 5202 \\ - 5432 \\ 112 - 4 \\ 0032 \end{matrix}], M = [\begin{matrix} 23 1 - 4 \\ 03 21 \\ - 53 42 \\ 2 3 - 51 \end{matrix}], N = [\begin{matrix} 2123 \\ 5254 \\ 2 - 2 - 13 \\ 4432 \end{matrix}], \end{matrix}

\begin{matrix} G = [\begin{matrix} - 5 522 \\ 1 - 123 \\ 4410 \\ - 2010 \end{matrix}], H = [\begin{matrix} - 5421 \\ 0325 \\ - 5261 \\ 2041 \end{matrix}], C = [\begin{matrix} 24 71 11 17 \\ 95 - 94 - 41 - 13 \\ 74 56 46 77 \\ 96 - 110 - 57 - 28 \end{matrix}], \\ D = [\begin{matrix} 103 - 13 38 110 \\ 100 - 94 - 54 - 64 \\ 32 - 67 - 113 - 20 \\ 45 21 4339 \end{matrix}], U = [\begin{matrix} 10 00 \\ 01 00 \\ 00 - 10 \\ 00 01 \end{matrix}], Z = [\begin{matrix} 1 00 0 \\ 0 - 10 0 \\ 001 0 \\ 000 - 1 \end{matrix}] \end{matrix}

It can be verified that matrix equations (1) are consistent over bisymmetric matrices and their bisymmetric solutions are as follows:

V = [\begin{matrix} 1 2 0 4 \\ 2 - 3 0 2 \\ 0 0 2 0 \\ 4 2 0 1 \end{matrix}] and W = [\begin{matrix} 2 0 3 0 \\ 0 3 0 - 2 \\ 3 0 3 0 \\ 0 - 2 0 2 \end{matrix}]

To obtain the matrix pair $[V, W]$ with $V \in BS R^{n \times n} (U)$ and $W \in BS R^{n \times n} (Z)$ such that Equations (1) are satisfied, we suppose that the initial matrices $V_{1} = 0$ and $W_{1} = 0$ . By applying Algorithm 2, we obtain the solutions as follows

\begin{matrix} V_{21} = [\begin{matrix} 1.0000 2.0000 - 0.00004.0000 \\ 2.0000 - 3.0000 0.00002.0000 \\ - 0.0000 - 0.0000 2.00000.0000 \\ 4.0000 2.0000 - 0.00001.0000 \end{matrix}], \\ W_{21} = [\begin{matrix} 2.0000 - 0.0000 3.0000 0.0000 \\ - 0.0000 3.0000 0.0000 - 2.0000 \\ 3.0000 - 0.0000 3.0000 0.0000 \\ 0.0000 - 2.0000 - 0.00002.0000 \end{matrix}] \end{matrix}

with the corresponding residual

‖ R_{21} ‖ = ‖ diag (C - f (V_{21}, W_{21}), D - g (V_{21}, W_{21})) ‖ = 2.3462 \times 1 0^{- 13}

The obtained results are presented in Table 5 and Figure 5.

Table 5.

The residual and the relative error for Example 3 with $V_{1} = W_{1} = 0$ .

Iteration number k	Residual r_k	Relative error δ_k
17	1.6516×10⁻⁷	3.2847×10⁻¹⁰
18	5.5115×10⁻⁹	1.6061×10⁻¹¹
19	6.3076×10⁻¹⁰	6.8180×10⁻¹³
20	2.2664×10⁻¹⁰	2.3688×10⁻¹¹
21	2.3462×10⁻¹³	1.7061×10⁻¹⁵

Figure 5.

The residual and the relative error for Example 3 with $V_{1} = W_{1} = 0$ .

Now let

\hat{V} = [\begin{matrix} 1 & 2 & 0 & 2 \\ 2 & - 2 & 0 & 2 \\ 0 & 0 & 2 & 0 \\ 2 & 2 & 0 & 1 \end{matrix}], \hat{W} = [\begin{matrix} 2 & 0 & 2 & 0 \\ 0 & 2 & 0 & - 2 \\ 2 & 0 & 2 & 0 \\ 0 & - 2 & 0 & 2 \end{matrix}]

By applying Algorithm 2 for the generalized coupled Sylvester matrix equations (31), and letting the initial pair $[{\bar{V}}_{1}, {\bar{W}}_{1}] = 0$ , we can obtain the least Frobenius norm generalized solution pair $[{\bar{V}}^{*}, {\bar{W}}^{*}]$ of the generalized coupled Sylvester matrix equations (31) as follows

\begin{matrix} {\bar{V}}^{*} = {\bar{V}}_{31} = [\begin{matrix} 0.0000 - 0.0000 0.00002.0000 \\ - 0.0000 - 1.0000 - 0.00000.0000 \\ 0.0000 - 0.0000 - 0.00000.0000 \\ 2.0000 - 0.0000 0.00000.0000 \end{matrix}], \\ {\bar{W}}^{*} = {\bar{W}}_{31} = [\begin{matrix} - 0.0000 0.0000 1.0000 0.0000 \\ - 0.00001.0000 - 0.0000 0.0000 \\ 1.0000 0.0000 1.0000 0.0000 \\ 0.0000 0.0000 - 0.0000 - 0.0000 \end{matrix}] \end{matrix}

with the corresponding residual

‖ R_{21} ‖ = ‖ diag (\bar{C} - f ({\bar{V}}_{21}, {\bar{W}}_{21}), \bar{D} - g ({\bar{V}}_{21}, {\bar{W}}_{21})) ‖ = 7.6968 \times 1 0^{- 14}

The obtained results are presented in Table 6 and Figure 6.

Table 6.

The residual and the relative error for Example 3 with ${\bar{V}}_{1} = {\bar{W}}_{1} = 0$ .

Iteration number k	Residual r_k	Relative error δ_k
17	3.1606×10⁻⁸	2.2496×10⁻¹⁰
18	1.5999×10⁻⁹	1.3480×10⁻¹¹
19	8.2791×10⁻¹⁰	5.3864×10⁻¹²
20	1.1910×10⁻¹⁰	1.1148×10⁻¹²
21	7.6968×10⁻¹⁴	1.0024×10⁻¹⁵

Figure 6.

The residual and the relative error for Example 2 with ${\bar{V}}_{1} = {\bar{W}}_{1} = 0$ .

Once the matrix pair $[{\bar{V}}^{*}, {\bar{W}}^{*}]$ is obtained, the solution $[\tilde{V}, \tilde{W}]$ of the nearness Problem 4 can be computed as follows

\begin{matrix} \tilde{V} = {\bar{V}}^{*} + \hat{V} = [\begin{matrix} 1.0000 2.0000 0.00004.0000 \\ 2.0000 - 3.0000 - 0.00002.0000 \\ 0.0000 - 0.0000 2.00000.0000 \\ 4.0000 2.0000 0.00001.0000 \end{matrix}], \\ \tilde{W} = {\bar{W}}^{*} + \hat{W} = [\begin{matrix} 2.0000 0.0000 3.0000 0.0000 \\ - 0.0000 3.0000 - 0.0000 - 2.0000 \\ 3.0000 0.0000 3.0000 0.0000 \\ 0.0000 - 2.0000 - 0.0000 2.0000 \end{matrix}] \end{matrix}

with the corresponding residual

‖ R_{21} ‖ = ‖ diag (C - f (\tilde{V}, \tilde{W}), D - g (\tilde{V}, \tilde{W})) ‖ = 8.0546 \times 1 0^{- 14}

Conclusion

This paper was concerned with the generalized bisymmetric solution to the generalized coupled Sylvester matrix equations. We presented three iterative algorithms to solve the generalized coupled Sylvester matrix equations. Firstly, we introduced a generalized solution to solve the generalized coupled Sylvester matrix equations; then a generalized bisymmetric (symmetric) solution was also constructed. With the iterative algorithms, the solvability of the generalized coupled Sylvester matrix equations can be determined automatically. Also, when the generalized coupled Sylvester matrix equations are consistent, we proved that for any initial matrix pair $[V_{1}, W_{1}]$ , generalized and generalized bisymmetric (symmetric) solutions can be obtained within a finite iteration steps. Also, the least Frobenius norm generalized solution pair in these cases can be obtained by choosing a suitable initial matrix pair. In addition, the optimal approximation solution to given generalized coupled Sylvester matrix equations can be derived by first finding the least Frobenius norm generalized solution pair of new generalized coupled Sylvester matrix equations then recover the optimal approximation solution of the nearness problem via certain relations. Numerical examples are given to support the theoretical results of this paper and to illustrate the accuracy of the proposed algorithms. The numerical results show also that the proposed method may be applied to solve more general forms of generalized coupled Sylvester matrix equations of over generalized bisymmetric matrices.

Footnotes

Appendix

Acknowledgements

The authors wish to express their sincere thanks to the anonymous referees for their helpful comments and suggestions.

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