An iterative algorithm to solve the generalized coupled Sylvester-transpose matrix equations

Abstract

This note presents an iterative algorithm to solve the coupled Sylvester-transpose matrix equations (including the generalized coupled Sylvester matrix equations and Lyapunov matrix equations as special cases) over generalized centro-symmetric matrices. When the considered matrix equations are consistent, for any initial generalized centro-symmetric matrix group, a generalized centro-symmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm generalized centro-symmetric solution group of the coupled Sylvester-transpose matrix equations can be derived when a suitable initial generalized centro-symmetric matrix group is chosen. In addition, for a given generalized centro-symmetric matrix group, the optimal approximation generalized centro-symmetric solution group can be obtained by finding the least Frobenius norm generalized centro-symmetric solution group of new coupled Sylvester-transpose matrix equations. Finally, a numerical example is given to demonstrate the efficiency of the introduced iterative algorithm.

Keywords

coupled Sylvester-transpose matrix equations least Frobenius norm generalized centro-symmetric solution iterative algorithm optimal approximation generalized centro-symmetric solution

Introduction

In this paper, we use the following notation. We use A ^T and ∥A∥ to denote transpose and the Frobenius norm of A, respectively. Here A⊗B = (a_ijB) means the Kronecker product of two matrices A and B. For two integers m < n, the symbol I[m, n] is used to represent the set {m, m+1, …, n}. Let R^m ^×n be the set of all m×n real matrices and SORⁿ ^×n be the set of all symmetric orthogonal matrices in Rⁿ ^×n. For a matrix $X = [\begin{matrix} x_{1} & x_{2} & \dots & x_{n} \end{matrix}] \in R^{m \times n}$ , $vec (X) = {[\begin{matrix} x_{1}^{T} & x_{2}^{T} & \dots & x_{n}^{T} \end{matrix}]}^{T}$ is the column stretching operation of X. The generalized centro-symmetric matrices can be defined as follows.

Definition 1.1. For arbitrary given matrix Q ∈ SORⁿ ^×n, i.e. Q = Q ^T = Q ⁻¹, if QAQ = A, the matrix A ∈ Rⁿ ^×n is called a generalized centro-symmetric matrix with respect to Q. The set of order n generalized centro-symmetric matrices with respect to Q is denoted by ${CSR}_{Q}^{n \times n}$ .

The generalized centro-symmetric matrix plays an important role in information theory, linear system theory, linear estimate theory and numerical analysis (Liang et al., 2007; Xie et al., 2006; Zhou et al., 2003). Chen (1998) puts forward the applications of the generalized centro-symmetric matrix in three major areas: the altitude estimation of a level network, an electric network and structural analysis of trusses. Coupled matrix equations have numerous applications in stability theory, control theory, perturbation analysis, signal processing and image restoration. For example, in stability analysis of linear jump systems with Markovian transitions, the continuous-time coupled Lyapunov matrix equations

A_{i}^{T} P_{i} + P_{i} A_{i} + Q_{i} + \sum_{j = 1}^{N} p_{ij} P_{j} = 0, Q_{i} > 0, i \in I [1, N]

and

P_{i} = A_{i}^{T} (\sum_{j = 1}^{N} p_{ij} P_{j}) A_{i} + Q_{i}, Q_{i} > 0, i \in I [1, N]

are often encountered (Borno, 1995; Mariton, 1990), where P_i > 0, i ∈ I[1, N] are the matrices to be determined. The coupled Sylvester matrix equations arise in computing error bounds for eigenvalue and eigenvalue space of the generalized eigenvalue problem (Kágström and Westin, 1989)

S - λ T = (\begin{matrix} A & - C \\ 0 & B \end{matrix}) - λ (\begin{matrix} D & - F \\ 0 & E \end{matrix})

in computing deflating subspace of the same problem, and in computing certain decomposition of the transferable matrix arising in control theory.

A large number of papers have studied several systems and matrix equations (Dehghan and Hajarian, 2008; Peng et al., 2006; Robbé and Sadkane, 2008; Shen and Chen, 2010). We often need to solve matrix equations which are shown in Table 1, where X, Y with size m×n are unknown matrices and A, B, C, D, E, F are arbitrary matrices of appropriate dimensions with real entries. But with the development of economy, science and technology, many researchers pay close attention to transpose matrix equation and conjugate matrix equation, which are shown in Table 2. The Sylvester-transpose matrix equation can be used to solve many control problems such as pole assignment (Dai, 1989), eigenstructure assignment (Fletcher et al., 1986) and robust fault detection (Frank, 1990). Recently, in Lin (2006), block Krylov subspace method was presented for solving CT Sylvester matrix equations. In Bao et al. (2007), a minimal residual method was reported for solving CT Sylvester matrix equations and DT Sylvester matrix equations. In Dehghan and Hajarian (2009), based on a global Arnoldi algorithm, a new projection method was presented for solving the large Sylvester matrix equation. In Wang et al. (2002), a finite iterative algorithm was constructed for solving the second-order Sylvester matrix equation EVF ²−AVF−CV = BW. In Wang (2005a,b) and Al Zhour and Kilicman (2007), Wang investigated the centro-symmetric solution to the matrix equation systems A ₁ X = C ₁, A ₃ XB ₃ = C ₃. Moreover, some necessary and sufficient solvability conditions, parameterized general solution and the maximal and minimal ranks of the general solution to the mixed Sylvester matrix equations and coupled Sylvester matrix equation were investigated (Wang and He, 2012, 2013a,b, 2014). The generalized Sylvester matrix equation, quaternion matrix equation and operator matrix equation have been studied by Wang et al. (2007a,b, 2009), Wang (2005a,b), and Wang and Wu (2010).

Table 1.

One-sided and two-sided Sylvester matrix equations.

Name	Matrix equation	Acronym
Continuous-time (CT) Sylvester	AX−XB = C	CTSY
Discrete-time (DT) Sylvester	X−AXB = C	DTSY
Generalized Sylvester	AXB−CXD = E	GSY
Coupled Sylvester	$\begin{matrix} AX - YB = C \\ DX - YE = F \end{matrix}$	CSY

Table 2.

Sylvester-transpose and Sylvester-conjugate matrix equations.

Name	Matrix equation	Acronym
Continuous-time (CT) Sylvester-transpose	AX−X ^T B = C	CTSYT
Discrete-time (DT) Sylvester-transpose	X−AX ^T B = C	DTSYT
Generalized Sylvester-transpose	AXB−CX ^T D = E	GSYT
Coupled Sylvester-transpose	$\sum_{j = 1}^{p} (A_{ij} X_{j} B_{ij} + C_{ij} X_{j}^{T} D_{ij}) = F_{i}, i \in I [1, N]$	CSYT
Continuous-time (CT) Sylvester-conjugate	$AX - \bar{X} B = C$	CTSYC
Discrete-time (DT) Sylvester-conjugate	$X - A \bar{X} B = C$	DTSYC
Generalized Sylvester-conjugate	$AXB - C \bar{X} D = E$	GSYC
Coupled Sylvester-conjugate	$\sum_{j = 1}^{p} (A_{ij} X_{j} B_{ij} + C_{ij} {\bar{X}}_{j} D_{ij}) = F_{i}, i \in I [1, N]$	CSYC

By applying the so-called generalized Kronecker map which had some elegant properties, Wu et al. (2012) discussed the closed-form solution to the generalized Sylvester-conjugate matrix equation. Zhou and Duan (2005) proposed gradient-based iterative algorithm for solving general coupled Sylvester matrix equations arising in linear systems theory. By using the so-called generalized Sylvester mapping, right coprime factorization and Bezout identity associated with certain polynomial matrix, Zhou et al. (2010) investigated the problem of parameterizing all solutions to the polynomial Diophantine matrix equation and the generalized Sylvester matrix equation. It is shown that the provided solutions can be parameterized as soon as two pairs of polynomial matrices satisfying the right coprime factorization and Bezout identity are obtained. Based on the conjugate gradient searching principle and its dual form, Li et al. (2010) presented two algorithms for solving the minimal norm least squares solution to a general linear equation $\sum_{i = 1}^{r} A_{i} {XB}_{i} + \sum_{j = 1}^{s} C_{j} X^{T} D_{j} = E$ . Between these two methods, the first is to minimize the spectral radius of the iteration matrix and explicit expression for the optimal step size is obtained. The second one is to minimize the square sum of the F-norm of the error matrices produced by the algorithm. The solvability, existence of unique solution, closed-form solution and numerical solution of matrix equation X = Af(X)B+C with $f (X) = X^{T}, f (X) = \bar{X}$ and f(X) = X^H , where X was the unknown matrix, were investigated (Zhou et al., 2011). In the work of Xie et al. (2009), the following matrix equation

\sum_{i = 1}^{r} A_{i} {XB}_{i} + \sum_{j = 1}^{s} C_{j} X^{T} D_{j} = E

(1.1)

was considered, where A_i , B_i , C_j , D_j , i = 1, …, r, j = 1, …, s, and E were some known constant matrices of appropriate dimensions and X was a matrix to be determined. In the work of Wang et al. (2007a), the solution of the generalized Sylvester-transpose matrix equation AXB+CX ^T D = E was presented by the iterative algorithm. The Moore–Penrose generalized inverse was used by Piao et al. (2007) to study the matrix equation AX+X ^T B = C and the explicit solutions to this matrix equation have been given. In the present paper, we propose an efficient iterative algorithm to solve the generalized centro-symmetric solution of the following coupled matrix equations

\sum_{j = 1}^{p} (A_{ij} X_{j} B_{ij} + C_{ij} X_{j}^{T} D_{ij}) = F_{i}, i \in I [1, N]

(1.2)

where $A_{ij} \in R^{m_{i} \times n_{j}}$ , $B_{ij} \in R^{n_{j} \times p_{i}}$ , $C_{ij} \in R^{m_{i} \times n_{j}}$ , $D_{ij} \in R^{n_{j} \times p_{i}}$ , $F_{i} \in R^{m_{i} \times p_{i}}$ , i ∈ I[1, N],j ∈ I[1, p], are the given known matrices, and $X_{j} \in R^{n_{j} \times n_{j}}, j \in I [1, p]$ are the matrices to be determined. It is known that the general coupled Sylvester-transpose matrix equations (1.2) are quite general and include the matrix equations in Li et al. (2010), Xie et al. (2009), Wang et al. (2007a,b) and Piao et al. (2007) as special cases. In this paper, we consider the following two problems of the coupled Sylvester-transpose matrix equations.

Problem 1.1. Given matrices $A_{ij} \in R^{m_{i} \times n_{j}}$ , $B_{ij} \in R^{n_{j} \times p_{i}}$ , $C_{ij} \in R^{m_{i} \times n_{j}}$ , $D_{ij} \in R^{n_{j} \times p_{i}}$ , $F_{i} \in R^{m_{i} \times p_{i}}$ , and symmetric orthogonal matrices $Q_{j} \in {SOR}^{n_{j} \times n_{j}}$ , i ∈ I[1, N], j ∈ I[1, p], find the generalized centro-symmetric matrix group (X ₁, X ₂, …, X_p ) with $X_{j} \in {CSR}^{n_{j} \times n_{j}}, j \in I [1, p]$ such that

\sum_{j = 1}^{p} (A_{ij} X_{j} B_{ij} + C_{ij} X_{j}^{T} D_{ij}) = F_{i}, i = 1, 2, \dots, N

(1.3)

in which the unknown matrices are satisfied X_j = Q_jX_jQ_j , j = 1,2,…,p.

Problem 1.2. Let Problem 1.1 be consistent, and its solution group set be denoted by S_r . For a given generalized centro-symmetric matrix group $({\hat{X}}_{1}, {\hat{X}}_{2}, \dots, {\hat{X}}_{p})$ with ${\hat{X}}_{j} \in {CSR}^{n_{j} \times n_{j}}, j \in I [1, p]$ , find $({\tilde{X}}_{1}, {\tilde{X}}_{2}, \dots, {\tilde{X}}_{p}) \in S_{r}$ with ${\tilde{X}}_{j} \in {CSR}^{n_{j} \times n_{j}}, j \in I [1, p]$ such that

\sum_{j = 1}^{p} ‖ {\tilde{X}}_{j} - {\hat{X}}_{j} ‖^{2} = min_{(X_{1}, \dots, X_{p}) \in S_{r}} {\sum_{j = 1}^{p} ‖ X_{j} - {\hat{X}}_{j} ‖^{2}}

(1.4)

The rest of this paper is outlined as follows. In Section 2, we first propose an efficient iterative method for deriving the generalized centro-symmetric solution to the coupled Sylvester-transpose matrix equation, then we obtain some properties of this iterative algorithm. It is proven that a generalized centro-symmetric solution group (the least Frobenius norm generalized centro-symmetric solution group) can be obtained by this algorithm for any (special) initial generalized centro-symmetric matrix group (X ₁(1), X ₂(1), …, X_p (1)) within finite steps in the absence of roundoff errors. By finding the least Frobenius norm generalized centro-symmetric solution group of corresponding new coupled Sylvester-transpose matrix equations, the optimal Problem 1.2 is solved. In Section 3, some numerical examples demonstrate that the proposed iterative algorithm is quite efficient. The conclusion can be found in Section 4.

Main result

In this section, we consider the following coupled Sylvester-transpose matrix equations

\sum_{j = 1}^{p} (A_{ij} X_{j} B_{ij} + C_{ij} X_{j}^{T} D_{ij}) = F_{i}, i \in I [1, N]

(2.1)

When the above matrix equations are consistent, we propose the following iterative algorithm.

Algorithm 2.1.

Step 1. Input matrices $A_{ij} \in R^{m_{i} \times n_{j}}$ , $B_{ij} \in R^{n_{j} \times p_{i}}$ , $C_{ij} \in R^{m_{i} \times n_{j}}$ , $D_{ij} \in R^{n_{j} \times p_{i}}$ , $F_{i} \in R^{m_{i} \times n_{i}}$ , the symmetric orthogonal matrices $Q_{j} \in {SOR}^{n_{j} \times n_{j}}$ , i ∈ I[1, N], and an arbitrary $X_{j} (1) \in {CSR}_{Q_{j}}^{n_{j} \times n_{j}}$ , j ∈ I[1,p]; Calculate

\begin{matrix} R_{i} (1) = F_{i} - \sum_{j = 1}^{p} [A_{ij} X_{j} (1) B_{ij} + C_{ij} X_{j}^{T} (1) D_{ij}], i \in I [1, N], \\ P_{j} (1) = \sum_{i = 1}^{N} \frac{1}{2} [A_{ij}^{T} R_{i} (1) B_{ij}^{T} + D_{ij} R_{i}^{T} (1) C_{ij} \\ + Q_{j} A_{ij}^{T} R_{i} (1) B_{ij}^{T} Q_{j} + Q_{j} D_{ij} R_{i}^{T} (1) C_{ij} Q_{j}], j \in I [1, p] . \end{matrix}

Step 2. If R_i (k) = 0, i ∈ I[1, N], then stop; otherwise, k = k+1.

Step 3. Calculate

X_{j} (k + 1) = X_{j} (k) + \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (k) ‖^{2}}{\sum_{τ = 1}^{p} ‖ P_{τ} (k) ‖^{2}} P_{j} (k), j \in I [1, p],

\begin{matrix} R_{i} (k + 1) & = F_{i} - \sum_{j = 1}^{p} [A_{ij} X_{j} (k + 1) B_{ij} + C_{ij} X_{j}^{T} (k + 1) D_{ij}] \\ = R_{i} (k) - \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (k) ‖^{2}}{\sum_{τ = 1}^{p} ‖ P_{τ} (k) ‖^{2}} \\ \sum_{j = 1}^{p} [A_{ij} P_{j} (k) B_{ij} + C_{ij} P_{j}^{T} (k) D_{ij}], i \in I [1, N], \end{matrix}

\begin{matrix} P_{j} (k + 1) = & \sum_{i = 1}^{N} \frac{1}{2} [A_{ij}^{T} R_{i} (k + 1) B_{ij}^{T} + D_{ij} R_{i}^{T} (k + 1) C_{ij} \\ + Q_{j} A_{ij}^{T} R_{i} (k + 1) B_{ij}^{T} Q_{j} \\ + Q_{j} D_{ij} R_{i}^{T} (k + 1) C_{ij} Q_{j}] \\ + \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (k + 1) ‖^{2}}{\sum_{ω = 1}^{N} ‖ R_{ω} (k) ‖^{2}} P_{j} (k), j \in I [1, p] . \end{matrix}

Step 4. Go to Step 2.

Remark 2.1. X_j (k) and P_j (k) generated by Algorithm 2.1 are the generalized centro-symmetric matrices with respect to Q_j , i.e.

X_{j} (k) = Q_{j} X_{j} (k) Q_{j}

and

P_{j} (k) = P_{j}^{T} (k) = Q_{j} P_{j} (k) Q_{j}

for k = 1,2,… and j = 1,2,…,p.

Now we introduce some properties of the above algorithm.

Lemma 2.1. Suppose that the sequences P_j (k), j ∈ I[1,p] and R_i (k),i ∈ I[1,N], are generated by Algorithm 2.1, then the following relation holds

\begin{matrix} \sum_{i = 1}^{N} t r [R_{i}^{T} (k + 1) R_{i} (m)] \\ = \sum_{i = 1}^{N} t r [R_{i}^{T} (k) R_{i} (m)] - \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (k) ‖^{2}}{\sum_{τ = 1}^{p} ‖ P_{τ} (k) ‖^{2}} \sum_{j = 1}^{p} t r [P_{j}^{T} (k) P_{j} (m)] \\ + \frac{[\sum_{ω = 1}^{N} ‖ R_{ω} (k) ‖^{2}] [\sum_{ω = 1}^{N} ‖ R_{ω} (m) ‖^{2}]}{[\sum_{τ = 1}^{p} ‖ P_{τ} (k) ‖^{2}] [\sum_{ω = 1}^{N} ‖ R_{ω} (m - 1) ‖^{2}]} \sum_{j = 1}^{p} t r [P_{j}^{T} (k) P_{j} (m - 1)] \end{matrix}

Proof. Because all matrices in Algorithm 2.1 are real, we can write

\begin{matrix} \sum_{i = 1}^{N} t r [R_{i}^{T} (k + 1) R_{i} (m)] \\ = \sum_{i = 1}^{N} t r [(R_{i} (k) - \frac{\sum_{ω = 1}^{N} {‖ R_{ω} (k) ‖}^{2}}{\sum_{τ = 1}^{p} {‖ P_{τ} (k) ‖}^{2}} \\ \sum_{j = 1}^{p} {(A_{ij} P_{j} (k) B_{ij} + C_{ij} P_{j}^{T} (k) D_{ij}))}^{T} R_{i} (m)] \\ = \sum_{i = 1}^{N} t r [R_{i}^{T} (k) R_{i} (m)] \\ - \frac{\sum_{ω = 1}^{N} {‖ R_{ω} (k) ‖}^{2}}{\sum_{τ = 1}^{p} ‖ P_{τ} (k) ‖^{2}} \sum_{i = 1}^{N} t r [\sum_{j = 1}^{p} (B_{ij}^{T} P_{j}^{T} (k) A_{ij}^{T} + D_{ij}^{T} P_{j} (k) C_{ij}^{T}) R_{i} (m)] \end{matrix}

By changing order of this double sum, we can obtain

\begin{matrix} \sum_{i = 1}^{N} t r [\sum_{j = 1}^{p} (B_{ij}^{T} P_{j}^{T} (k) A_{ij}^{T} + D_{ij}^{T} P_{j} (k) C_{ij}^{T}) R_{i} (m)] \\ = \sum_{j = 1}^{p} t r [\sum_{i = 1}^{N} (P_{j}^{T} (k) A_{ij}^{T} R_{i} (m) B_{ij}^{T} + P_{j} (k) C_{ij}^{T} R_{i} (m) D_{ij}^{T})] \\ = \sum_{j = 1}^{p} t r [P_{j}^{T} (k) (\sum_{i = 1}^{N} A_{ij}^{T} R_{i} (m) B_{ij}^{T}) + P_{j} (k) (\sum_{i = 1}^{N} C_{ij}^{T} R_{i} (m) D_{ij}^{T})] \\ = \sum_{j = 1}^{p} t r [P_{j}^{T} (k) (\sum_{i = 1}^{N} A_{ij}^{T} R_{i} (m) B_{ij}^{T}) + P_{j}^{T} (k) (\sum_{i = 1}^{N} D_{ij} R_{i}^{T} (m) C_{ij})] \end{matrix}

\begin{matrix} = \sum_{j = 1}^{p} t r [P_{j}^{T} (k) \sum_{i = 1}^{N} (A_{ij}^{T} R_{i} (m) B_{ij}^{T} + D_{ij} R_{i}^{T} (m) C_{ij})] \\ = \sum_{j = 1}^{p} t r [P_{j}^{T} (k) (P_{j} (m) - \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (m) ‖^{2}}{\sum_{ω = 1}^{N} ‖ R_{ω} (m - 1) ‖^{2}} P_{j} (m - 1))] \\ = \sum_{j = 1}^{p} t r [P_{j}^{T} (k) P_{j} (m)] - \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (j) ‖^{2}}{\sum_{ω = 1}^{N} ‖ R_{ω} (m - 1) ‖^{2}} \sum_{j = 1}^{p} t r [P_{j}^{T} (k) P_{j} (m - 1)] . \end{matrix}

Combining the preceding two relations, gives the conclusion of this lemma.▪

Lemma 2.2. Suppose that the sequences {P_j (k)}, j ∈ I[1, p] and {R_i (k)}, i ∈ I[1, N] are generated by Algorithm 2.1. If there exists an integer number φ ≥ 1, such that $\sum_{i = 1}^{N} {| R_{i} (k) ‖}^{2} \neq 0$ , for all k = 1,…,φ, then

\sum_{i = 1}^{N} t r [R_{i}^{T} (m) R_{i} (n)] = 0, \sum_{j = 1}^{p} t r [P_{j}^{T} (m) P_{j} (n)] = 0

for all m, n = 1,…,φ, m≠n.

Proof. It is obvious that

\begin{matrix} tr [R_{i}^{T} (m) R_{i} (n)] = tr [R_{i}^{T} (n) R_{i} (m)], \\ tr [P_{j}^{T} (m) P_{j} (n)] = tr [P_{j}^{T} (n) P_{j} (m)] \end{matrix}

We only need to show

\sum_{i = 1}^{N} t r [R_{i}^{T} (m) R_{i} (n)] = 0, \sum_{j = 1}^{p} t r [P_{j}^{T} (m) P_{j} (n)] = 0

(2.2)

for 1 ≤ n < m ≤ φ.

In the following, we use induction to prove conclusion (2.2) and we do it in two steps.

Step 1. First of all, we have

\begin{matrix} \sum_{i = 1}^{N} t r [R_{i}^{T} (2) R_{i} (1)] = \sum_{i = 1}^{N} t r \\ [{(R_{i} (1) - \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (1) ‖^{2}}{\sum_{τ = 1}^{p} ‖ P_{τ} (1) ‖^{2}} \sum_{j = 1}^{p} (A_{ij} P_{j} (1) B_{ij} + C_{ij} P_{j}^{T} (1) D_{ij})}^{T} R_{i} (1)] \\ = \sum_{i = 1}^{N} t r [R_{i}^{T} (1) R_{i} (1)] - \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (1) ‖^{2}}{\sum_{τ = 1}^{p} ‖ P_{τ} (1) ‖^{2}} \\ \sum_{i = 1}^{N} t r [\sum_{j = 1}^{p} (B_{ij}^{T} P_{j}^{T} (1) A_{ij}^{T} + D_{ij}^{T} P_{j} (1) C_{ij}^{T}) R_{i} (1)] \\ = \sum_{i = 1}^{N} ‖ R_{i} (1) ‖^{2} - \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (1) ‖^{2}}{\sum_{τ = 1}^{p} ‖ P_{τ} (1) ‖^{2}} \sum_{i = 1}^{N} t r \\ [\sum_{j = 1}^{p} P_{j}^{T} (1) (A_{ij}^{T} R_{i} (1) B_{ij}^{T} + D_{ij} R_{i}^{T} (1) C_{ij})] \\ = \sum_{i = 1}^{N} ‖ R_{i} (1) ‖^{2} - \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (1) ‖^{2}}{\sum_{τ = 1}^{p} ‖ P_{τ} (1) ‖^{2}} tr \\ [\sum_{j = 1}^{p} P_{j}^{T} (1) (\frac{\sum_{i = 1}^{N} (A_{ij}^{T} R_{i} (1) B_{ij}^{T} + D_{ij} R_{i}^{T} (1) C_{ij})}{2} \\ + \frac{\sum_{i = 1}^{N} Q_{j} (A_{ij}^{T} R_{i} (1) B_{ij}^{T} + D_{ij} R_{i}^{T} (1) C_{ij}) Q_{j}}{2})] \end{matrix}

\begin{matrix} = \sum_{i = 1}^{N} ‖ R_{i} (1) ‖^{2} - \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (1) ‖^{2}}{\sum_{τ = 1}^{p} ‖ P_{τ} (1) ‖^{2}} tr [\sum_{j = 1}^{p} P_{j}^{T} (1) P_{j} (1)] \\ = \sum_{i = 1}^{N} ‖ R_{i} (1) ‖^{2} - \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (1) ‖^{2}}{\sum_{τ = 1}^{p} ‖ P_{τ} (1) ‖^{2}} \sum_{j = 1}^{p} ‖ P_{j} (1) ‖^{2} = 0 \end{matrix}

According to Algorithm 2.1, we can obtain

Assume that (2.2) holds for n = v−1. For n = v, it follows that

\begin{matrix} \sum_{i = 1}^{N} t r [R_{i}^{T} (v + 1) R_{i} (v)] \\ = \sum_{i = 1}^{N} t r [(R_{i} (v) - \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (v) ‖^{2}}{\sum_{τ = 1}^{p} ‖ P_{τ} (v) ‖^{2}} \\ \sum_{j = 1}^{p} {(A_{ij} P_{j} (v) B_{ij} + C_{ij} P_{j}^{T} (v) D_{ij})}^{T} R_{i} (v)] \\ = \sum_{i = 1}^{N} t r [R_{i}^{T} (v) R_{i} (v)] - \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (v) ‖^{2}}{\sum_{τ = 1}^{p} ‖ P_{τ} (v) ‖^{2}} \\ \sum_{i = 1}^{N} t r [\sum_{j = 1}^{p} (B_{ij}^{T} P_{j}^{T} (v) A_{ij}^{T} + D_{ij}^{T} P_{j} (v) C_{ij}^{T}) R_{i} (v)] \\ = \sum_{i = 1}^{N} ‖ R_{i} (v) ‖^{2} - \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (v) ‖^{2}}{\sum_{τ = 1}^{p} ‖ P_{τ} (v) ‖^{2}} \\ \sum_{i = 1}^{N} t r [\sum_{j = 1}^{p} P_{j}^{T} (v) (A_{ij}^{T} R_{i} (v) B_{ij}^{T} + D_{ij} R_{i}^{T} (v) C_{ij})] \\ = \sum_{i = 1}^{N} ‖ R_{i} (v) ‖^{2} - \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (v) ‖^{2}}{\sum_{τ = 1}^{p} ‖ P_{τ} (v) ‖^{2}} tr \\ [\sum_{j = 1}^{p} P_{j}^{T} (v) (\frac{\sum_{i = 1}^{N} (A_{ij}^{T} R_{i} (v) B_{ij}^{T} + D_{ij} R_{i}^{T} (v) C_{ij})}{2} \\ + \frac{\sum_{i = 1}^{N} Q_{j} (A_{ij}^{T} R_{i} (v) B_{ij}^{T} + D_{ij} R_{i}^{T} (v) C_{ij}) Q_{j}}{2})] \\ = \sum_{i = 1}^{N} ‖ R_{i} (v) ‖^{2} - \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (v) ‖^{2}}{\sum_{τ = 1}^{p} ‖ P_{τ} (v) ‖^{2}} tr \\ [\sum_{j = 1}^{p} P_{j}^{T} (v) (P_{j} (v) - \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (v) ‖^{2}}{\sum_{ω = 1}^{N} ‖ R_{ω} (v - 1) ‖^{2}} P_{j} (v - 1))] \\ = \sum_{i = 1}^{N} ‖ R_{i} (v) ‖^{2} - \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (v) ‖^{2}}{\sum_{τ = 1}^{p} ‖ P_{τ} (v) ‖^{2}} \sum_{j = 1}^{p} ‖ P_{j} (v) ‖^{2} = 0 \end{matrix}

In addition, we have

Hence, by the principle of induction, the conclusion (2.2) holds for all n = 1, 2, …, s.

Step 2. In this step, it is assumed that

\sum_{i = 1}^{N} t r [R_{i}^{T} (n + m) R_{i} (n)] = 0, \sum_{j = 1}^{p} t r [P_{j}^{T} (n + m) P_{j} (n)] = 0

for 1 ≤ n ≤ s, 1 < m < s.

Now we prove

\begin{matrix} \sum_{i = 1}^{N} t r [R_{i}^{T} (n + m + 1) R_{i} (n)] = 0, \\ \sum_{j = 1}^{p} t r [P_{j}^{T} (n + m + 1) P_{j} (n)] = 0, \end{matrix}

for 1 ≤ n ≤ s, 1 < m < s.

By using the previous results, we can obtain

where C ₁ is a certain known matrix.

Moreover, we have

\begin{matrix} \sum_{j = 1}^{p} t r [P_{j}^{T} (n + m + 1) P_{j} (n)] \\ = \sum_{j = 1}^{p} t r [(\frac{\sum_{i = 1}^{N} (A_{ij}^{T} R_{i} (n + m + 1) B_{ij}^{T} + D_{ij} R_{i}^{T} (n + m + 1) C_{ij})}{2} \\ + \frac{\sum_{i = 1}^{N} Q_{j} (A_{ij}^{T} R_{i} (n + m + 1) B_{ij}^{T} + D_{ij} R_{i}^{T} (n + m + 1) C_{ij}) Q_{j}}{2} \\ {+ \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (n + m + 1) ‖^{2}}{\sum_{ω = 1}^{N} ‖ R_{ω} (n + m) ‖^{2}} P_{j} (n + m))}^{T} P_{j} (n)] \\ = \sum_{j = 1}^{p} t r [P_{j}^{T} (n) (\sum_{i = 1}^{N} (A_{ij}^{T} R_{i} (n + m + 1) B_{ij}^{T} + D_{ij} R_{i}^{T} (n + m + 1) C_{ij})] \\ + \frac{\sum_{ω = 1}^{N} ‖ R_{ω} (n + m + 1) ‖^{2}}{\sum_{ω = 1}^{N} ‖ R_{ω} (n + m) ‖^{2}} \sum_{j = 1}^{p} t r [P_{j}^{T} (n + m) P_{j} (n)] = 0 \end{matrix}

Thus, the second relation in (2.2) holds for 1 ≤ n ≤ s, 1 < m < s.

From steps 1 and 2 above, the conclusion is immediately obtained by the principle of induction.▪

Lemma 2.3. Suppose that the coupled Sylvester-transpose matrix equation (2.1) are consistent, and [X _1*, X _2*, …, X_p _*] is a generalized centro-symmetric solution group. Then, for any initial generalized centro-symmetric matrices X_n (1), n ∈ I[1, p], the sequences {X_j (k)}, {P_j (k)}, j ∈ I[1, p] and {R_i (k)}, i ∈ I[1, N], generated by Algorithm 2.1 satisfy

\sum_{j = 1}^{p} t r [P_{j}^{T} (k) (X_{j *} - X_{j} (k))] = \sum_{i = 1}^{N} ‖ R_{i} (k) ‖^{2}, k = 1, 2, \dots

(2.3)

Proof. We prove the conclusion by induction. For k = 1, we have

\begin{matrix} \sum_{j = 1}^{p} t r [P_{j}^{T} (1) (X_{j *} - X_{j} (1))] \\ = \sum_{j = 1}^{p} t r [{(\frac{\sum_{i = 1}^{N} (A_{ij}^{T} R_{i} (1) B_{ij}^{T} + D_{ij} R_{i}^{T} (1) C_{ij})}{2} + \frac{\sum_{i = 1}^{N} Q_{j} (A_{ij}^{T} R_{i} (1) B_{ij}^{T} + D_{ij} R_{i}^{T} (1) C_{ij}) Q_{j}}{2})}^{T} (X_{j *} - X_{j} (1))] \\ = \sum_{j = 1}^{p} t r [\sum_{i = 1}^{N} {(A_{ij}^{T} R_{i} (1) B_{ij}^{T} + D_{ij} R_{i}^{T} (1) C_{ij})}^{T} (X_{j *} - X_{j} (1))] \\ = \sum_{j = 1}^{p} t r [\sum_{i = 1}^{N} (B_{ij} R_{i}^{T} (1) A_{ij} + C_{ij}^{T} R_{i} (1) D_{ij}^{T}) (X_{j *} - X_{j} (1))] \\ = \sum_{j = 1}^{p} t r [\sum_{i = 1}^{N} (R_{i}^{T} (1) A_{ij} (X_{j *} - X_{j} (1)) B_{ij} + R_{i} (1) D_{ij}^{T} (X_{j *} - X_{j} (1)) C_{ij}^{T}] \\ = \sum_{j = 1}^{p} t r [\sum_{i = 1}^{N} (R_{i}^{T} (1) A_{ij} (X_{j *} - X_{j} (1)) B_{ij} + R_{i}^{T} (1) C_{ij} {(X_{j *} - X_{j} (1))}^{T} D_{ij})] \\ = \sum_{i = 1}^{N} t r [R_{i}^{T} (1) \sum_{j = 1}^{p} (A_{ij} (X_{j *} - X_{j} (1)) B_{ij} + C_{ij} {(X_{j *} - X_{j} (1))}^{T} D_{ij})] \\ = \sum_{i = 1}^{N} t r [R_{i}^{T} (1) (F_{i} - \sum_{j = 1}^{p} (A_{ij} X_{j} (1) B_{ij} + C_{ij} X_{j}^{T} (1) D_{ij}))] \\ = \sum_{i = 1}^{N} t r [R_{i}^{T} (1) R_{i} (1)] = \sum_{i = 1}^{N} {‖ R_{i} (1) ‖}^{2} \end{matrix}

Now assume the conclusion (2.3) holds for k = n. Then, we can get

\sum_{j = 1}^{p} t r [P_{j}^{T} (n) (X_{j *} - X_{j} (n))] = \sum_{i = 1}^{N} ‖ R_{i} (n) ‖^{2}

By Algorithm 2.1, we can obtain

\begin{matrix} \sum_{j = 1}^{p} t r [P_{j}^{T} (n + 1) (X_{j *} - X_{j} (n + 1))] \\ = \sum_{j = 1}^{p} t r [(\frac{\sum_{i = 1}^{N} (A_{ij}^{T} R_{i} (n + 1) B_{ij}^{T} + D_{ij} R_{i}^{T} (n + 1) C_{ij})}{2} \\ {+ \frac{\sum_{i = 1}^{N} Q_{j} (A_{ij}^{T} R_{i} (n + 1) B_{ij}^{T} + D_{ij} R_{i}^{T} (n + 1) C_{ij}) Q_{j}}{2})}^{T} (X_{j *} - X_{j} (n + 1))] \\ = \sum_{j = 1}^{p} t r [\sum_{i = 1}^{N} {(A_{ij}^{T} R_{i} (n + 1) B_{ij}^{T} + D_{ij} R_{i}^{T} (n + 1) C_{ij})}^{T} (X_{j *} - X_{j} (n + 1))] \\ = \sum_{j = 1}^{p} t r [\sum_{i = 1}^{N} (B_{ij} R_{i}^{T} (n + 1) A_{ij} + C_{ij}^{T} R_{i} (n + 1) D_{ij}^{T}) (X_{j *} - X_{j} (n + 1))] \end{matrix}

In addition, we have

\begin{matrix} \sum_{j = 1}^{p} t r [\sum_{i = 1}^{N} (B_{ij} R_{i}^{T} (n + 1) A_{ij} + C_{ij}^{T} R_{i} (n + 1) D_{ij}^{T}) (X_{j *} - X_{j} (n + 1))] \\ = \sum_{j = 1}^{p} t r [\sum_{i = 1}^{N} (R_{i}^{T} (n + 1) A_{ij} (X_{j *} - X_{j} (n + 1)) B_{ij} + R_{i} (n + 1) D_{ij}^{T} (X_{j *} - X_{j} (n + 1)) C_{ij}^{T})] \\ = \sum_{j = 1}^{p} t r [\sum_{i = 1}^{N} (R_{i}^{T} (n + 1) A_{ij} (X_{j *} - X_{j} (n + 1)) B_{ij} + R_{i}^{T} (n + 1) C_{ij} {(X_{j *} - X_{j} (n + 1))}^{T} D_{ij})] \\ = \sum_{i = 1}^{N} t r [\sum_{j = 1}^{p} R_{i}^{T} (n + 1) (A_{ij} (X_{j *} - X_{j} (n + 1)) B_{ij} + C_{ij} {(X_{j *} - X_{j} (n + 1))}^{T} D_{ij})] \\ = \sum_{i = 1}^{N} t r [R_{i}^{T} (n + 1) (F_{i} - \sum_{j = 1}^{p} (A_{ij} X_{j} (n + 1) B_{ij} + C_{ij} X_{j}^{T} (n + 1) D_{ij})] \\ = \sum_{i = 1}^{N} ‖ R_{i} (n + 1) ‖^{2} \end{matrix}

So we have

\sum_{j = 1}^{p} t r [P_{j}^{T} (n + 1) (X_{j *} - X_{j} (n + 1))] = \sum_{i = 1}^{N} ‖ R_{i} (n + 1) ‖^{2}

This implies that (2.3) holds for k = n+1, which completes the proof.▪

Theorem 2.1. Assume that Problem 1.1 is consistent, then for any arbitrary initial matrix group [X ₁, X ₂, …, X_p ] with $X_{j} (1) \in {CSR}_{Q_{j}}^{n_{j} \times n_{j}}$ , j = 1,2,…,p, a generalized centro-symmetric solution group of Problem 1.1 can be obtained within finite iteration steps in the absence of roundoff errors.

Proof. First, in the space $R^{m_{1} \times p_{1}} \times R^{m_{2} \times p_{2}} \times \dots \times R^{m_{N} \times p_{N}}$ , we define an inner product as

〈 (A_{1}, A_{2}, \dots, A_{N}), (B_{1}, B_{2}, \dots, B_{N}) 〉 = \sum_{i = 1}^{N} t r (B_{i}^{T} A_{i})

for $(A_{1}, A_{2}, \dots, A_{N}), (B_{1}, B_{2}, \dots, B_{N}) \in R^{m_{1} \times p_{1}} \times R^{m_{2} \times p_{2}} \times \dots \times R^{m_{N} \times p_{N}}$ .

If $\sum_{i = 1}^{N} ‖ R_{i} (k) ‖^{2} \neq 0, k = 1, 2, \dots, q = 2 \sum_{i = 1}^{N} m_{i} p_{i}$ , then from previous lemmas we can conclude $\sum_{j = 1}^{p} ‖ P_{j} (k) ‖^{2} \neq 0$ . Now we can compute R_i (q+1) and [X ₁(q+1), X ₂(q+1), …, X_p (q+1)]. From Lemma 2.2, it is not difficult to get

\sum_{i = 1}^{N} (R_{i}^{T} (q + 1) R_{i} (j)) = 0, j = 1, 2, \dots, q

and

\sum_{i = 1}^{N} (R_{i}^{T} (m) R_{i} (j)) = 0, m, j = 1, 2, \dots, q, m \neq j

Then [R ₁(k), R ₂(k), …, R_N (k)], k = 1, 2, …, 2q, is a group of orthogonal basis of the previously defined inner product space. It follows that [R ₁(q+1), R ₂(q+1), …, R_N (q+1)] = 0 and [X ₁(q+1), X ₂(q+1),…, X_p (q+1)] = 0 is a solution group of Problem 1.1. Therefore, when Problem 1.1 is consistent, we can verify that the solution group of Problem 1.1 can be obtained within finite iterative steps.▪

Lemma 2.4 (Al Zhour and Kilicman, 2007). Let P(m, n) ∈ R^mn ^×mn be a square mn×mn matrix partitioned into m×n sub-matrices such that ijth position 1 and zero elsewhere, i.e.

P (m, n) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} E_{ij} \otimes E_{ij}^{T}

where $E_{ij} = e_{i} e_{j}^{T}$ called an elementary matrix of order m×1(n×1). Using this definition, we have

vec (X^{T}) = P (m, n) vec (X), P (m, n) P (n, m) = I_{mn}, P^{T} (m, n) = P^{- 1} (m, n) = P (n, m)

Lemma 2.5 (Ben-Israel and Greville, 2003). Suppose that the consistent system of linear equation Ax = b has a solution x* ∈ R(A^T ), then x* is the unique least norm solution to the systems of linear equation.

Theorem 2.2. Assume that the coupled Sylvester-transpose matrix equations (2.1) are consistent, then, if we take the initial matrices

\begin{matrix} X_{i} (1) = \sum_{i = 1}^{N} (A_{ji}^{T} E_{j} B_{ji}^{T} + D_{ji} E_{j}^{T} C_{ji}) + \sum_{j = 1}^{N} Q_{j} (A_{ji}^{T} E_{j} B_{ji}^{T} + D_{ji} E_{j}^{T} C_{ji}) Q_{j}, i = 1, 2, \dots, p \end{matrix}

where E_j , j = 1, 2, …, N, are arbitrary matrices, or especially X_i (1) = 0, then the solution group $(X_{1 *}, X_{2 *}, \dots, X_{p *})$ generated by Algorithm 2.1 is the least Frobenius norm generalized centro-symmetric solution group of the coupled Sylvester-transpose matrix equations (2.1).

Proof. Let E_j be arbitrary matrices for j = 1, 2, …, N,

M_{1} = (\begin{matrix} Vec (\sum_{j = 1}^{N} (A_{j 1}^{T} E_{j} B_{j 1}^{T} + D_{j 1} E_{j}^{T} C_{j 1})) \\ Vec (\sum_{j = 1}^{N} (A_{j 2}^{T} E_{j} B_{j 2}^{T} + D_{j 2} E_{j}^{T} C_{j 2})) \\ ⋮ \\ Vec (\sum_{j = 1}^{N} (A_{jp}^{T} E_{j} B_{jp}^{T} + D_{jp} E_{j}^{T} C_{jp})) \end{matrix})

M_{2} = (\begin{matrix} Vec (\sum_{j = 1}^{N} Q_{1} (A_{j 1}^{T} E_{j} B_{j 1}^{T} + D_{j 1} E_{j}^{T} C_{j 1}) Q_{1}) \\ Vec (\sum_{j = 1}^{N} Q_{2} ((A_{j 2}^{T} E_{j} B_{j 2}^{T} + D_{j 2} E_{j}^{T} C_{j 2}) Q_{2}) \\ ⋮ \\ Vec (\sum_{j = 1}^{N} Q_{N} (A_{jp}^{T} E_{j} B_{jp}^{T} + D_{jp} E_{j}^{T} C_{jp}) Q_{N}) \end{matrix})

N_{1} = (\begin{matrix} B_{11} \otimes A_{11}^{T} + (C_{11}^{T} \otimes D_{11}) P (m_{1}, p_{1}) & \dots & B_{N 1} \otimes A_{N 1}^{T} + (C_{N 1}^{T} \otimes D_{N 1}) P (m_{N}, p_{N}) \\ B_{12} \otimes A_{12}^{T} + (C_{12}^{T} \otimes D_{12}) P (m_{1}, p_{1}) & \dots & B_{N 2} \otimes A_{N 2}^{T} + (C_{N 2}^{T} \otimes D_{N 2}) P (m_{N}, p_{N}) \\ ⋮ & ⋮ & ⋮ \\ B_{1 p} \otimes A_{1 p}^{T} + (C_{1 p}^{T} \otimes D_{1 p}) P (m_{1}, p_{1}) & \dots & B_{Np} \otimes A_{Np}^{T} + (C_{Np}^{T} \otimes D_{Np}) P (m_{N}, p_{N}) \end{matrix})

\begin{matrix} N_{2} = (\begin{matrix} Q_{1} B_{11} \otimes Q_{1} A_{11}^{T} + (Q_{1} C_{11}^{T} \otimes Q_{1} D_{11}) P (m_{1}, p_{1}) & \dots & \dots \\ Q_{2} B_{12} \otimes Q_{2} A_{12}^{T} + (Q_{2} C_{12}^{T} \otimes Q_{2} D_{12}) P (m_{1}, p_{1}) & \dots & \dots \\ ⋮ & ⋮ & ⋮ \\ Q_{p} B_{1 p} \otimes Q_{p} A_{1 p}^{T} + (Q_{p} C_{1 p}^{T} \otimes Q_{p} D_{1 p}) P (m_{1}, p_{1}) & \dots & \dots \end{matrix} \\ \begin{matrix} \dots & \dots & Q_{1} B_{N 1} \otimes Q_{1} A_{N 1}^{T} + (Q_{1} C_{N 1}^{T} \otimes Q_{1} D_{N 1}) P (m_{N}, p_{N}) \\ \dots & \dots & Q_{2} B_{N 2} \otimes Q_{2} A_{N 2}^{T} + (Q_{2} C_{N 2}^{T} \otimes Q_{2} D_{N 2}) P (m_{N}, p_{N}) \\ ⋮ & ⋮ & ⋮ \\ \dots & \dots & Q_{p} B_{Np} \otimes Q_{p} A_{Np}^{T} + (Q_{p} C_{Np}^{T} \otimes Q_{p} D_{Np}) P (m_{N}, p_{N}) \end{matrix}) \\ M_{1} = N_{1} (\begin{matrix} Vec (E_{1}) \\ Vec (E_{2}) \\ ⋮ \\ Vec (E_{N}) \end{matrix}), M_{2} = N_{2} (\begin{matrix} Vec (E_{1}) \\ Vec (E_{2}) \\ ⋮ \\ Vec (E_{N}) \end{matrix}) \end{matrix}

So we have

\begin{matrix} M_{1} + M_{2} = (\begin{matrix} N_{1} & N_{2} \end{matrix}) (\begin{matrix} Vec (E_{1}) \\ ⋮ \\ Vec (E_{N}) \\ Vec (E_{1}) \\ ⋮ \\ Vec (E_{N}) \end{matrix}) \\ = {(\begin{matrix} N_{1}^{T} \\ N_{2}^{T} \end{matrix})}^{T} (\begin{matrix} Vec (E_{1}) \\ ⋮ \\ Vec (E_{N}) \\ Vec (E_{1}) \\ ⋮ \\ Vec (E_{N}) \end{matrix}) \in R ({(\begin{matrix} N_{1}^{T} \\ N_{2}^{T} \end{matrix})}^{T}) \end{matrix}

Now if we consider the initial matrices

\begin{matrix} X_{i} (1) = \sum_{j = 1}^{N} (A_{ji}^{T} E_{j} B_{ji}^{T} + D_{ji} E_{j}^{T} C_{ji}) + \sum_{j = 1}^{N} Q_{j} (A_{ji}^{T} E_{j} B_{ji}^{T} + D_{ji} E_{j}^{T} C_{ji}) Q_{j}, i = 1, 2, \dots, p \end{matrix}

where E_j , j = 1, 2, …, N, are the arbitrary matrices, then all X_i (k) for i = 1, 2, …, p, generated by Algorithm 2.1 satisfy

(\begin{matrix} Vec (X_{1} (k)) \\ Vec (X_{2} (k)) \\ ⋮ \\ Vec (X_{p} (k)) \end{matrix}) \in R ({(\begin{matrix} N_{1}^{T} \\ N_{2}^{T} \end{matrix})}^{T})

Hence, if we choose the initial matrix group (X ₁(1), X ₂(1), …, X_p (1)) where

\begin{matrix} X_{i} (1) = & \sum_{j = 1}^{N} (A_{ji}^{T} E_{j} B_{ji}^{T} + D_{ji} E_{j}^{T} C_{ji}) \\ + \sum_{j = 1}^{N} Q_{j} (A_{ji}^{T} E_{j} B_{ji}^{T} + D_{ji} E_{j}^{T} C_{ji}) Q_{j}, i = 1, 2, \dots, p \end{matrix}

according to Lemma 2.5, the solution group $(X_{1 *} (1), X_{2 *} (1), \dots, X_{p *} (1))$ obtained by Algorithm 2.1 is the least Frobenius norm solution group.▪

In the following, we will solve Problem 1.2.

When the matrix equations (2.1) are consistent, the solution group set S_E of the coupled matrix equations (2.1) are non-empty. For a given matrix group $({\hat{X}}_{1}, {\hat{X}}_{2}, \dots, {\hat{X}}_{p})$ with ${\hat{X}}_{j} \in R^{l_{j} \times n_{j}}, j = 1, 2, \dots, p,$ it is not difficult to get

\begin{matrix} \sum_{j = 1}^{N} (A_{ij} X_{j} B_{ij} + C_{ij} X_{j}^{T} D_{ij}) = F_{i}, i = 1, 2, \dots, N \\ \Leftrightarrow \sum_{j = 1}^{N} (A_{ij} (X_{j} - {\hat{X}}_{j}) B_{ij} + C_{ij} {(X_{j} - {\hat{X}}_{j})}^{T} D_{ij}) \\ = F_{i} - \sum_{j = 1}^{N} (A_{ij} {\hat{X}}_{j} B_{ij} + C_{ij} {\hat{X}}_{j}^{T} D_{ij}), i = 1, 2, \dots, N \end{matrix}

Define ${\bar{X}}_{j} = X_{j} - {\hat{X}}_{j}$ and ${\bar{F}}_{i} = F_{i} - \sum_{j = 1}^{N} (A_{ij} {\hat{X}}_{j} B_{ij} + C_{ij} {\hat{X}}_{j}^{T} D_{ij})$ for i = 1, 2, …, N, then the matrix nearness Problem 1.2 is equivalent to find the least Frobenius norm solution group $({\bar{X}}_{1 *}, {\bar{X}}_{2 *}, \dots, {\bar{X}}_{p *})$ of new coupled Sylvester-transpose matrix equations

\sum_{j = 1}^{N} (A_{ij} {\bar{X}}_{j} B_{ij} + C_{ij} {\bar{X}}_{j}^{T} D_{ij}) = {\bar{F}}_{i}, i = 1, 2, \dots, N

That can be obtained by using Algorithm 2.1 with the initial matrices

\begin{matrix} {\bar{X}}_{i} (1) = \sum_{j = 1}^{N} (A_{ji}^{T} E_{j} B_{ji}^{T} + D_{ji} E_{j}^{T} C_{ji}) \\ + \sum_{j = 1}^{N} Q_{j} (A_{ji}^{T} E_{j} B_{ji}^{T} + D_{ji} E_{j}^{T} C_{ji}) Q_{j}, i = 1, 2, \dots, p \end{matrix}

where E_j are arbitrary matrices, or especially, ${\bar{X}}_{i} (1) = 0$ for i = 1, 2, …, p. Here the solution group of the matrix nearness Problem 1.2 can be represented as $({\tilde{X}}_{1}, {\tilde{X}}_{2}, \dots, {\tilde{X}}_{p})$ with

{\tilde{X}}_{j} = {\bar{X}}_{j *} + {\hat{X}}_{j}, j = 1, 2, \dots, p

Numerical example

In this section, we report a numerical example to test the proposed iterative method.

Example 3.1. Consider the following coupled Sylvester-transpose matrix equations

{\begin{matrix} A_{11} {XB}_{11} + C_{11} X^{T} D_{11} + A_{12} {YB}_{12} + C_{12} Y^{T} D_{12} = E_{1} \\ A_{21} {XB}_{21} + C_{21} X^{T} D_{21} + C_{22} Y^{T} D_{22} = E_{2} \end{matrix}

(3.1)

with the following parametric matrices

\begin{matrix} A_{11} = (\begin{matrix} - 1 & 2 & 1 & 1 & 1 \\ 2 & 2 & 3 & 1 & 0 \\ 2 & 1 & 2 & 2 & 3 \\ 1 & 2 & 5 & 1 & 1 \\ 1 & 2 & 0 & 2 & 3 \end{matrix}), B_{11} = (\begin{matrix} 1 & 2 & 1 & 3 & 3 \\ 5 & 2 & 4 & 1 & 0 \\ 1 & 1 & 1 & 4 & 0 \\ 2 & 6 & 2 & 1 & 1 \\ 2 & 1 & 3 & 4 & 6 \end{matrix}), \\ A_{12} = (\begin{matrix} 1 & 4 & 2 & 6 & 0 \\ 2 & 2 & 6 & 1 & 0 \\ 3 & 2 & 1 & 6 & 11 \\ 2 & 3 & 5 & 2 & 4 \\ 2 & 3 & 0 & 1 & 2 \end{matrix}) \end{matrix}

\begin{matrix} A_{21} = (\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 4 & 2 \\ 1 & 5 & 3 & 1 & 3 \\ 1 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 \end{matrix}), B_{21} = (\begin{matrix} 4 & 3 & 2 & 2 & 3 \\ 4 & 1 & 2 & 1 & 4 \\ 2 & 3 & 3 & 1 & 5 \\ 4 & 2 & 2 & 0 & 0 \\ 2 & 2 & 2 & 1 & 1 \end{matrix}), \\ C_{21} = (\begin{matrix} 21 & - 2 & 11 & 1 & 0 \\ 2 & - 2 & - 3 & 1 & 0 \\ 2 & 13 & 2 & 2 & 1 \\ 11 & 32 & 5 & 1 & 1 \\ 1 & 2 & 2 & 1 & 4 \end{matrix}) \end{matrix}

\begin{matrix} D_{21} = (\begin{matrix} 1 & 22 & 12 & 53 \\ 5 & 2 & - 4 & 1 \\ - 1 & 1 & - 1 & 4 \\ - 2 & 6 & 2 & 1 \end{matrix}), \\ C_{22} = (\begin{matrix} 0 & 2 & 0 & 1 & 4 \\ 0 & 11 & 21 & 1 & 5 \\ 4 & 0 & 2 & 1 & 5 \\ 0 & - 1 & - 1 & 1 & - 3 \\ 1 & 2 & 3 & - 1 & 2 \end{matrix}), \\ D_{22} = (\begin{matrix} - 3 & 12 & 11 & 2 & 2 \\ 21 & 23 & 1 & 2 & 3 \\ 1 & 2 & 4 & 1 & 5 \\ 2 & 11 & 3 & 2 & 5 \\ 0 & - 1 & - 2 & 11 & 4 \end{matrix}) \end{matrix}

\begin{matrix} E_{1} = (\begin{matrix} 517 & 1686 & 1805 & 1962 & 666 \\ 324 & 161 & 133 & 376 & 138 \\ 275 & 399 & 477 & 232 & 484 \\ 344 & 346 & 362 & 263 & 462 \\ 150 & 210 & 252 & 46 & 249 \end{matrix}), \\ E_{2} = (\begin{matrix} 185 & 910 & 278 & 1320 & 272 \\ 994 & 1064 & 272 & 681 & 589 \\ 272 & 1119 & 502 & 1259 & 345 \\ - 230 & 1547 & 779 & 2473 & 347 \\ 194 & 463 & 139 & 611 & 169 \end{matrix}) \end{matrix}

B_{12} = (\begin{matrix} 2 & 1 & 3 & 1 & 1 \\ 4 & 3 & 2 & 6 & 4 \\ 4 & 2 & 3 & 1 & 4 \\ 1 & 2 & 3 & 0 & 5 \\ 2 & 3 & 0 & 3 & 4 \end{matrix}), C_{11} = (\begin{matrix} 0 & 2 & 1 & 1 & 0 \\ 0 & - 5 & 3 & 1 & 0 \\ 6 & 1 & 7 & 2 & 3 \\ 2 & 2 & 4 & 1 & 3 \\ 3 & 2 & 2 & 1 & 4 \end{matrix})

\begin{matrix} D_{11} = (\begin{matrix} - 2 & 2 & 11 & 3 & 3 \\ - 2 & 2 & 4 & 1 & 3 \\ 1 & 1 & 0 & 0 & 4 \\ 2 & 6 & 2 & - 8 & - 1 \\ - 2 & - 1 & - 2 & 3 & 4 \end{matrix}), \\ C_{12} = (\begin{matrix} 13 & 2 & 14 & 61 & 1 \\ 2 & 12 & 0 & 1 & 0 \\ 2 & 1 & 2 & 0 & 0 \\ 1 & 2 & 6 & 1 & 1 \\ 1 & 2 & - 1 & - 2 & 3 \end{matrix}), \\ D_{12} = (\begin{matrix} 1 & 12 & 11 & 13 & 1 \\ 5 & 2 & 4 & 1 & 11 \\ 10 & 1 & 1 & 4 & 2 \\ - 2 & - 6 & 2 & 1 & 2 \\ 1 & 2 & 0 & 1 & 0 \end{matrix}) \end{matrix}

It can be verified that the generalized coupled matrix equations (3.1) are consistent over generalized centro-symmetric matrices and have the generalized centro-symmetric solution pair (X*, Y*) with

\begin{matrix} X^{*} = (\begin{matrix} 1 & 1 & 0 & 2 & 1 \\ 1 & - 1 & 0 & 2 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 2 & 2 & 0 & 0 & 1 \\ 1 & 1 & 0 & 1 & 1 \end{matrix}) \in {CSR}_{Q_{1}}^{5 \times 5}, \\ Y^{*} = (\begin{matrix} 1 & 0 & 0 & 2 & 0 \\ 0 & - 1 & 2 & 0 & 1 \\ 0 & 2 & 1 & 0 & 1 \\ 2 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 \end{matrix}) \in {CSR}_{Q_{1}}^{5 \times 5} \end{matrix}

where

\begin{matrix} Q_{1} = (\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) \in {SOR}^{5 \times 5}, \\ Q_{2} = (\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & - 1 \end{matrix}) \in {SOR}^{5 \times 5} \end{matrix}

Taking the initial matrices (X(1), Y(1)) = 0 and applying Algorithm 2.1, we obtain the solutions

\begin{matrix} X (235) = (\begin{matrix} 1.0000 & 0.9999 & 0.0000 & 2.0000 & 0.9999 \\ 0.9999 & - 1.0000 & 0.0000 & 2.0000 & 1.0000 \\ 0.0000 & 0.0000 & 0.9999 & 0.0000 & 0.0000 \\ 1.9999 & 2.0000 & 0.0000 & 0.0000 & 1.0000 \\ 1.0000 & 0.9999 & 0.0000 & 0.9999 & 0.9999 \end{matrix}) \\ Y (235) = (\begin{matrix} 0.9999 & 0.0000 & 0.0000 & 2.0000 & 0.0000 \\ 0.0000 & - 1.0000 & 2.0000 & 0.0000 & 0.9999 \\ 0.0000 & 2.0000 & 0.9999 & 0.0000 & 1.0000 \\ 2.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 1.0000 & 0.9999 & 0.0000 & 0.9999 \end{matrix}) \end{matrix}

with corresponding residual

‖ R (235) ‖ = 8.4111 \times 10^{- 11}

The obtained results are presented in Figure 1, where

e_{k} = \frac{‖ (X (k), Y (k) - (X^{*}, Y^{*}) ‖}{‖ (X^{*}, X^{*}) ‖}, r_{k} = ‖ R (k) ‖

Figure 1.

The relative error of solution and the residual for Example 3.1.

Now let

\hat{X} = (\begin{matrix} 2 & 2 & 0 & 2 & 2 \\ 2 & 2 & 0 & 2 & 2 \\ 0 & 0 & 2 & 0 & 0 \\ 2 & 2 & 0 & 2 & 2 \\ 2 & 2 & 0 & 2 & 2 \end{matrix}), \hat{Y} = (\begin{matrix} 2 & 0 & 0 & 2 & 0 \\ 0 & 2 & 2 & 0 & 2 \\ 0 & 2 & 2 & 0 & 2 \\ 2 & 0 & 0 & 2 & 0 \\ 0 & 2 & 2 & 0 & 2 \end{matrix})

By Algorithm 2.1 for the generalized coupled Sylvester matrix equations (3.1), choosing the initial generalized centro-symmetric matrix pair $(\bar{X} (1), \bar{Y} (1)) = 0$ , we have the least Frobenius norm generalized centro-symmetric solutions of the generalized coupled Sylvester matrix equations (3.1) with the following form:

\begin{matrix} {\bar{X}}^{*} = \bar{X} (231) = (\begin{matrix} - 0.9999 & - 1.0000 & 0.0000 & 0.0000 & - 1.0000 \\ - 1.0000 & - 2.9999 & 0.0000 & 0.0000 & - 0.9999 \\ 0.0000 & 0.0000 & - 0.9999 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & - 1.9999 & - 0.9999 \\ - 0.9999 & - 1.0000 & 0.0000 & 0.9999 & - 1.0000 \end{matrix}) \\ {\bar{Y}}^{*} = \bar{Y} (231) (\begin{matrix} - 0.9999 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & - 3.0000 & 0.0000 & 0.0000 & - 0.9999 \\ 0.0000 & 0.0000 & - 0.9999 & 0.0000 & - 1.0000 \\ 0.0000 & 0.0000 & 0.0000 & - 2.0000 & 0.0000 \\ 0.0000 & - 0.9999 & - 1.0000 & 0.0000 & - 1.0000 \end{matrix}) \end{matrix}

with corresponding residual

‖ R (231) ‖ = 7.1420 \times 10^{- 11}

The obtained results are presented in Figure 2. Therefore, the solutions of the matrix equation nearness problem are

\begin{matrix} \tilde{X} = {\bar{X}}^{*} + \hat{X} = (\begin{matrix} 1.0001 & 1.0000 & 0.0000 & 2.0000 & 1.0000 \\ 1.0000 & - 0.9999 & 0.0000 & 2.0000 & 1.0001 \\ 0.0000 & 0.0000 & 1.0001 & 0.0000 & 0.0000 \\ 2.0000 & 2.0000 & 0.0000 & 0.0001 & 1.0001 \\ 1.0001 & 1.0000 & 0.0000 & 1.0001 & 1.0000 \end{matrix}) \\ \tilde{Y} = {\bar{Y}}^{*} + \hat{Y} = (\begin{matrix} 1.0001 & 0.0000 & 0.0000 & 2.0000 & 0.0000 \\ 0.0000 & - 1.0000 & 2.0000 & 0.0000 & 1.0001 \\ 0.0000 & 2.0000 & 1.0001 & 0.0000 & 1.0000 \\ 2.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 1.0001 & 1.0000 & 0.0000 & 1.0000 \end{matrix}) \end{matrix}

Figure 2.

The relative error of the least Frobenius norm generalized centro-symmetric solution and the residual for Example 3.1.

Conclusions

Many problems in control and system theory are solved by computing the solutions of coupled Sylvester matrix equations. Coupled matrix equations have numerous applications in stability theory, signal processing and image restoration. The generalized centro-symmetric matrices play an important role in information theory, the altitude estimation of a level network and structural analysis of trusses. Therefore, we propose an iterative algorithm for solving the generalized centro-symmetric solution group of the coupled Sylvester-transpose matrix equations. By this algorithm, the solvability of the generalized centro-symmetric solution group to the generalized coupled Sylvester-transpose matrix equations can be derived automatically. When the considered coupled matrix equations are consistent, for any initial generalized centro-symmetric matrix group, a generalized centro-symmetric solution group can be obtained in finite iterative steps in the absence of roundoff errors. Furthermore, the optimal approximation problem is solved by using the proposed Algorithm 2.1.

Footnotes

Acknowledgements

The authors are very grateful to the anonymous reviewers and the editor for their helpful comments and suggestions which have helped us in improving the quality of this paper.

Funding

This work was supported by the NNSF (grant number 61374025), Research Awards Young and Middle-Aged Scientists of Shandong Province (grant number BS2011SF009) and Excellent Youth Foundation of Shandong’s Natural Scientific Committee (grant number JQ201219).

References

Al Zhour

Kilicman

(2007) Some new connections between matrix products for partitioned and non-partitioned matrices. Computers and Mathematics with Applications 54(6): 763–784.

Bao

Lin

Wei

(2007) A new projection method for solving large Sylvester equations. Applied Numerical Mathematics 57: 521–532.

Ben-Israel

Greville

TNE

(2003) Generalized Inverse: Theory and Applications. New York: Springer.

Borno

(1995) Parallel computation of the solution of coupled algebric Lyapunov equations. Automatica 31(9): 1345–1347.

Chen

(1998) Generalized reflexive matrices: special properties and applicaitons. SIAM Journal on Matrix Analysis and Applications 19: 140–153.

Dai

(1989) Singular Control Systems. Berlin: Springer-Verlag.

Dehghan

Hajarian

(2008) An iterative algorithm for solving a pair of matrix equations AY B = E,CY D = F over generealized centro-symmetric matrices. Computers and Mathematics with Applications 56: 3246–3260.

Dehghan

Hajarian

(2009) Efficient iterative method for solving the second-order Sylvester matrix equation EV F ² − AVF − CV = BW. IET Control Theory Applications 3: 1401–1408.

Fletcher

Kuatsky

Nichols

(1986) Eigenstructure assignment in descriptor systems. IEEE Transactions on Automatic Control 31: 1138–1141.

10.

Frank

(1990) Fault diagnosis in dynamic systems using analytical and knowledgebased redundancy a survey and some new results. Automatica 26: 459–474.

11.

Kágström

Westin

(1989) Generalized Schur methods with condition estimators for solving the generalized Sylvester equation. IEEE Transactions on Automatic Control 34: 745–751.

12.

Wang

Zhou

Duan

(2010) Least squares solution with the minimum-norm to general matrix equations via iteration. Applied Mathematics and Computation 215: 3547–3562.

13.

Liang

You

Dai

(2007) An efficient algorithm for the generalized centro-symmetric solution of matrix equation AXB = C. Numerical Algorithms 4: 173–184.

14.

Lin

(2006) Minimal residual methods augmented with eigenvectors for solving Sylvester equations and generalized Sylvester equations. Applied Mathematics and Computation 181: 487–499.

15.

Mariton

(1990) Jump Linear Systems in Automatic Control New York: Marcel Dekker.

16.

Peng

Zhang

(2006) An iterative method for symmetric solutions and optimal approximation solution of the system of matrix equations A ₁ XB ₁ = C ₁, A ₂ XB ₂ = C ₂ . Applied Mathematics and Computation 183: 1127–1137.

17.

Piao

Zhang

Wang

(2007) The solution to matrix equation AX+X ^T C = B. Journal of the Franklin Institute 344: 1056–1062.

18.

Robbé

Sadkane

(2008) Use of near-breakdowns in the block Arnildi method for solving large Sylvester equations. Applied Numerical Mathematics 58(4): 486–498.

19.

Shen

Chen

(2010) An iterative method for the symmetric and skew symmetric solutions of a linear matrix equation AXB+CYD = E. Journal of Computatational and Applied Mathematics 233: 3030–3040.

20.

Wang

Cheng

Wei

(2007a) Iterative algorithms for solving the matrix equation AXB+CX ^T D = E. Applied Mathematics and Computation 187: 622–629.

21.

Wang

(2005a) Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations. Computers and Mathematics with Applications 49: 641–650.

22.

Wang

(2005b) A system of four matrix equations over von Neumann regular rings and its applications. Acta Mathematica Scientia 21(2): 323–334.

23.

Wang

(2012) Some matrix equations with applications. Linear and Multilinear Algebra 60(11–12): 1327–1353.

24.

Wang

(2013a) Solvability conditions and general solution for mixed Sylvester equations. Automatica 49: 2713–2719.

25.

Wang

(2013b) A system of matrix equations and its applications. Science China Mathematics 56(9): 1795–1820.

26.

Wang

(2014) Systems of coupled generalized Sylvester matrix equations. Automatica 50: 2840–2844.

27.

Wang

Song

Lin

(2007b) Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application. Applied Mathematics and Computation 189: 1517–1532.

28.

Wang

Sun

(2002) Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra. Linear Algebra and its Applications 353: 169–182.

29.

Wang

van der Woude

Chang

(2009) A system of real quaternion matrix equations with applications. Linear Algebra and its Applications 431: 2291–2303.

30.

Wang

(2010) Common Hermitian solutions to some operator equations on Hilbert C-modules. Linear Algebra and its Applications 432: 3159–3171.

31.

Zhan

Liu

(2012) On closed-form solutions to the generalized Sylvester-conjugate matrix equation. Applied Mathematics and Computation 218(19): 9730–9741.

32.

Xie

Sheng

(2006) The solvability conditions for the inverse eigenproblems of symmetric and generalized centro-symmetric matrices and their approximations. Linear Algebra Applications 418: 142–152.

33.

Xie

Ding

(2009) Gradient based iterative solutions for general linear matrix equations. Computers and Mathematics with Applications 58: 1441–1448.

34.

Zhou

Duan

(2005) An explicit solution to the matrix equation AX–XF=BY, Applied Mathematics Letters, 402: 345–366.

35.

Zhou

James

Lam

Duan

(2011) Toward solution of matrix equation X = Af(X)B+C. Linear Algebra and its Applications 435(6): 1370–1398.

36.

Zhou

Yan

Duan

(2010) Unified parametrization for the solutions to the polynomial Diophantine matrix equation and the generalized Sylvester matrix equation. International Journal of Control Automation and Systems 8(1): 29–35.

37.

Zhou

Zhang

(2003) The solvability conditions for the inverse eigen problem of generalized centro-symmetric matrices. Linear Algebra Applications 364: 147–160.