Gradient-based iterative algorithm for solving the generalized coupled Sylvester-transpose and conjugate matrix equations over reflexive (anti-reflexive) matrices

Abstract

Linear matrix equations play an important role in many areas, such as control theory, system theory, stability theory and some other fields of pure and applied mathematics. In the present paper, we consider the generalized coupled Sylvester-transpose and conjugate matrix equations

$T_{ν} (X) = F_{ν}, ν = 1, 2, \dots, N,$

where $X = (X_{1}, X_{2}, \dots, X_{p})$ is a group of unknown matrices and for $ν = 1, 2, \dots, N$ ,

$T_{ν} (X) = \sum_{i = 1}^{p} \sum_{μ = 1}^{s_{1}} A_{ν i μ} X_{i} B_{ν i μ} + \sum_{μ = 1}^{s_{2}} C_{ν i μ} X_{i}^{T} D_{ν i μ} + \sum_{μ = 1}^{s_{3}} M_{ν i μ} {\bar{X}}_{i} N_{ν i μ} + \sum_{μ = 1}^{s_{4}} H_{ν i μ} X_{i}^{H} G_{ν i μ},$

in which $A_{ν i μ}$ , $B_{ν i μ}$ , $C_{ν i μ}$ , $D_{ν i μ}$ , $M_{ν i μ}$ , $N_{ν i μ}$ , $H_{ν i μ}$ , $G_{ν i μ}$ and $F_{ν}$ are given matrices with suitable dimensions defined over complex number field. By using the hierarchical identification principle, an iterative algorithm is proposed for solving the above coupled linear matrix equations over the group of reflexive (anti-reflexive) matrices. Meanwhile, sufficient conditions are established which guarantee the convergence of the presented algorithm. Finally, some numerical examples are given to demonstrate the validity of our theoretical results and the efficiency of the algorithm for solving the mentioned coupled linear matrix equations.

Keywords

Generalized Sylvester-transpose and conjugate matrix equation iterative algorithm reflexive (anti-reflexive) matrix

Introduction

In this paper, the following notation is utilized. We use $tr (A), A^{T}, A^{H}, \bar{A}$ to denote the trace, the transpose, the conjugate transposed, the conjugate of the matrix $A$ , respectively. Moreover, $ℂ^{m \times n}$ represents the set of all $m \times n$ complex matrices. The set of all symmetric orthogonal matrices, also known as reflection matrices, in $ℂ^{n \times n}$ is represented by $S O ℂ^{n \times n}$ , i.e. $P \in S O ℂ^{n \times n}$ if and only if P = P^H = P⁻¹.

Definition 1.1 (Chen, 1998). Consider two arbitrary given matrices $P, Q \in S O ℂ^{n \times n}$ . The matrix $A \in ℂ^{n \times n}$ is called a reflexive matrix, with respect to $P$ and $Q$ , if $A = PAQ$ . The set of all $n \times n$ reflexive matrices, with respect to $P$ and $Q$ , is denoted by $ℂ_{r}^{n \times n} (P, Q) .$

Definition 1.2 (Chen, 1998). Consider two arbitrary given matrices $P, Q \in S O ℂ^{n \times n}$ . The matrix $A \in ℂ^{n \times n}$ is called an anti-reflexive matrix, with respect to $P$ and $Q$ , if $A = - PAQ$ . The set of all $n \times n$ anti-reflexive matrices, with respect to $P$ and $Q$ , is denoted by $ℂ_{a}^{n \times n} (P, Q) .$

In various research works, the hierarchical identification principle has been exploited to present different algorithms for specific problems. The idea has been originally utilized by Ding and Chen (2005a–e). For example, Ding and Chen (2005a) have obtained the lifted state-space models for general dual-rate systems with different updating and sampling periods. Furthermore, based on a hierarchical identification principle, estimation algorithms have been presented to identify the canonical lifted models based on the given dual-rate input–output data in which the causality constraints of the lifted systems have been taken into account. In another paper, Ding and Chen (2005b) have expanded a hierarchical least squares iterative (HLSI) algorithm and a hierarchical least squares (HLS) algorithm, invoking a hierarchical identification principle, for multivariable discrete-time systems described by transfer matrices. In an alternative paper, Ding and Chen (2005c) have propounded a hierarchical gradient iterative algorithm and a hierarchical stochastic gradient algorithm and established that the parameter estimation errors given by the algorithms converge to zero for any initial values under persistent excitation. Some of the recently published works, related to this subject, include Ding et al. (2011), Han et al. (2010), Liu et al. (2012), Wang et al. (2012) and Zhang et al. (2011).

In the literature, the problem of finding solutions of several linear matrix equations has been investigated widely, for more details see Beik and Salkuyeh (2011), Bouhamidi and Jbilou (2008), Chang and Wang (1993), Ding and Chen (2005d,e, 2006), Ding et al. (2008, 2010), Jbilou and Riquet (2006), Salkuyeh and Toutounian (2006), Song et al. (2011), Wu et al. (2011b), Zhang (2011), Zhou and Duan (2008) and the references therein. Before stating the main problems of this paper, we briefly review some of the works which have been recently presented in the field of linear matrix equations.

Recently, the idea of a conjugate gradient (CG) method has been developed for constructing iterative algorithms to compute the solutions of different kinds of linear matrix equations over reflexive and anti-reflexive, generalized bisymmetric, generalized centro-symmetric, mirror-symmetric, skew-symmetric and $(P, Q)$ -reflexive matrices, for more details see Dehghan and Hajarian (2008, 2010, 2011a), Hajarian and Dehghan (2011), Huang et al. (2008), Li et al. (2008, 2010a), Liang et al. (2007), Wang and Wu (2011) and Wu et al. (2011a).

The gradient-based iterative algorithm is a different common approach for solving linear matrix equations (for further details see Ding and Chen (2005d, 2006) and Ding et al. (2008, 2010) Ding and Chen (2005e) have presented a family of iterative methods for solving the coupled matrix equations by using hierarchical identification principle and the star product. More precisely, Ding et al. (2008) have considered the solution of $AXB = F$ and $AXB + CXD = F$ . Dehghan and Hajarian (2011b) have proposed two algorithms for solving the following the matrix equation

\sum_{i = 1}^{p} A_{i} X B_{i} + \sum_{j = 1}^{q} C_{j} Y D_{j} = F,

(1.1)

over reflexive and anti-reflexive matrices.

In an alternative paper, Dehghan and Hajarian (2012a) have employed the idea of the Gradient-based iterative method to construct two iterative algorithms for computing the generalized bisymmetric and skew-symmetric solutions of the linear matrix equation

\sum_{i = 1}^{l} A_{i} Y B_{i} = C .

(1.2)

Moreover, Song et al. (2011) have considered the following coupled Sylvester-transpose matrix equations

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

(1.3)

where $A_{i η} \in ℝ^{m_{i} \times l_{η}}$ , $B_{i η} \in ℝ^{n_{η} \times p_{i}}$ , $C_{i η} \in ℝ^{m_{i} \times n_{η}}$ , $D_{i η} \in ℝ^{l_{η} \times p_{i}}$ , $F_{i} \in ℝ^{m_{i} \times p_{i}}$ , $i = 1, \dots, N$ , $η = 1, \dots, p$ , are given matrices and $X_{η} \in ℝ^{l_{η} \times n_{η}}, η \in I [1, p]$ are the matrices to be determined.

More recently, a gradient-based iterative method has been proposed for solving the following coupled linear matrix equations

{\begin{matrix} A_{1} X B_{1} + C_{1} X^{T} D_{1} = M_{1}, \\ A_{2} X B_{2} + C_{2} X^{T} D_{2} = M_{2}, \end{matrix}

(1.4)

over reflexive and anti-reflexive matrices (Dehghan and Hajarian, 2012b).

In addition, Dehghan and Hajarian (2012c) have presented a gradient-based iterative method for solving the generalized coupled Sylvester matrix equations which contain (coupled) Sylvester and Lyapunov matrix equations as special cases.

For simplicity, we use the following linear operator

T_{ν} : ℂ^{p_{1} \times p_{1}} \times \dots \times ℂ^{p_{p} \times p_{p}} \to ℂ^{λ_{ν} \times γ_{ν}}

such that

\begin{matrix} T_{ν} (X) = \sum_{i = 1}^{p} (\sum_{μ = 1}^{s_{1}} A_{ν i μ} X_{i} B_{ν i μ} + \sum_{μ = 1}^{s_{2}} C_{ν i μ} X_{i}^{T} D_{ν i μ} \\ + \sum_{μ = 1}^{s_{3}} M_{ν i μ} {\bar{X}}_{i} N_{ν i μ} + \sum_{μ = 1}^{s_{4}} H_{ν i μ} X_{i}^{H} G_{ν i μ} \end{matrix}),

where $X = (X_{1}, X_{2}, \dots, X_{p})$ and $ν = 1, 2, \dots, N$ .

In the present work, we will construct a gradient-based algorithm for solving the following coupled linear matrix equations

T_{ν} (X) = F_{ν}, ν \in I [1, N],

(1.5)

over the group of reflexive (anti-reflexive) matrices. Evidently, the coupled linear matrix equations (1.5) are extremely general and include many investigated linear matrix equations such as the generalized (coupled) Sylvester and Lyapunov matrix equations, (1.1), (1.2), (1.3) and (1.4).

Problem reformulation

We will focus on the following problems and construct an iterative algorithm for obtaining the solutions of these problems.

Problem 1. Assume that the matrices $A_{ν i μ} \in ℂ^{λ_{ν} \times p_{i}}, B_{ν i μ} \in ℂ^{p_{i} \times γ_{ν}}$ , $μ \in I [1, s_{1}]$ , $C_{ν i μ} \in ℂ^{λ_{ν} \times p_{i}}, D_{ν i μ} \in ℂ^{p_{i} \times γ_{ν}}$ , $μ \in I [1, s_{2}]$ , $M_{ν i μ} \in ℂ^{λ_{ν} \times p_{i}}, N_{ν i μ} \in ℂ^{p_{i} \times γ_{ν}}$ , $μ \in I [1, s_{3}]$ , $H_{ν i μ} \in ℂ^{λ_{ν} \times p_{i}}, G_{ν i μ} \in ℂ^{p_{i} \times γ_{ν}}$ , $μ \in I [1, s_{4}]$ , $F_{ν} \in ℂ^{λ_{ν} \times γ_{ν}}$ and $R_{i}, Q_{i} \in S O ℂ^{p_{i} \times p_{i}}$ are given. Find the reflexive matrix group $(X_{1}, X_{2}, \dots, X_{p})$ such that $X_{i} \in ℂ_{r}^{p_{i} \times p_{i}} (R_{i}, Q_{i})$ and satisfy (1.5) where $i \in I [1, p], ν \in I [1, N]$ .

Problem 2. Assume that the matrices $A_{ν i μ} \in ℂ^{λ_{ν} \times p_{i}}, B_{ν i μ} \in ℂ^{p_{i} \times γ_{ν}}$ , $μ \in I [1, s_{1}]$ , $C_{ν i μ} \in ℂ^{λ_{ν} \times p_{i}}, D_{ν i μ} \in ℂ^{p_{i} \times γ_{ν}}$ , $μ \in I [1, s_{2}]$ , $M_{ν i μ} \in ℂ^{λ_{ν} \times p_{i}}, N_{ν i μ} \in ℂ^{p_{i} \times γ_{ν}}$ , $μ \in I [1, s_{3}]$ , $H_{ν i μ} \in ℂ^{λ_{ν} \times p_{i}}, G_{ν i μ} \in ℂ^{p_{i} \times γ_{ν}}$ , $μ \in I [1, s_{4}]$ , $F_{ν} \in ℂ^{λ_{ν} \times γ_{ν}}$ and $R_{i}, Q_{i} \in S O ℂ^{p_{i} \times p_{i}}$ are given. Find the anti-reflexive matrix group $(X_{1}, X_{2}, \dots, X_{p})$ such that $X_{i} \in ℂ_{a}^{p_{i} \times p_{i}} (R_{i}, Q_{i})$ and satisfy (1.5) where $i \in I [1, p], ν \in I [1, N]$ .

The reminder of this paper is organized as follows. In Section 2, we recall some definitions and theorems which are used for presenting our main theoretical results. In Section 3, first, we investigate the solvability of Problems 1 and 2. Then, an algorithm is proposed for solving these problems. Convergence analysis of the algorithm is also discussed. In order to illustrate the validity and applicability of our presented results, we give some numerical examples in Section 4. Finally, the paper is ended with a brief conclusion in Section 5.

Preliminaries

In this section, we review some necessary principles and definitions which are utilized throughout this work.

Definition 2.1. Suppose that $Y = (Y_{1}, Y_{2}, \dots, Y_{k})$ and $Z = (Z_{1}, Z_{2}, \dots, Z_{k})$ where $Y_{i}, Z_{i} \in ℂ^{r_{i} \times s_{i}}$ for $i = 1, 2, \dots, k$ . We define the inner product $〈 \cdot, \cdot 〉$ as follows:

〈 Y, Z 〉 = Re [\sum_{i = 1}^{k} tr (Y_{j}^{H} Z_{j})] .

(2.1)

Remark 2.2. For $Y = (Y_{1}, Y_{2}, \dots, Y_{k})$ , $Y_{i} \in ℂ^{r_{i} \times s_{i}}$ for $i \in I [1, k]$ , the norm of $Y$ is defined by $‖ Y ‖^{2} = Re [\sum_{i = 1}^{k} tr (Y_{i}^{H} Y_{i})] .$

Assume that $A = {[a_{ij}]}_{m \times s}$ and $B = {[b_{ij}]}_{n \times q}$ defined over complex (real) number field, the Kronecker product of the matrices $A$ and $B$ is defined as the $mn \times sq$ matrix $A \otimes B = [a_{ij} B] .$ The “ $vec$ ” operator transforms a matrix $A$ of size $m \times s$ to a vector $a = vec (A)$ of size $ms \times 1$ by stacking the columns of $A$ . In this paper, the following relation is utilized (Bernstein, 2009):

vec (AXB) = (B^{T} \otimes A) vec (X) .

Lemma 2.3. Let $X \in ℝ^{m \times n}$ be an arbitrary matrix. Then

vec (X^{T}) = P (m, n) vec (X),

where $P (m, n)$ is uniquely determined by the integers $m$ and $n$ .

Proof. See Chen and Chen (2002). □

Remark 2.4. Let $X \in ℂ^{m \times n}$ be an arbitrary matrix. Then

vec (X^{T}) = P (m, n) vec (X),

and

vec (X^{H}) = vec ({\bar{X}}^{T}) = P (m, n) vec (\bar{X}),

where $P (m, n)$ is uniquely determined by the integers $m$ and $n$ .

Some properties of the matrix $P (m, n)$ are given as follows (Li et al., 2010b):

For two arbitrary integers $m$ and $n$ , $P (m, n)$ has the following explicit form

P (m, n) = {(\begin{matrix} E_{11}^{T} & E_{12}^{T} & \dots & E_{1 n}^{T} \\ E_{21}^{T} & E_{22}^{T} & \dots & E_{2 n}^{T} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ E_{m 1}^{T} & E_{m 2}^{T} & \dots & E_{mn}^{T} \end{matrix})}_{mn \times mn} .

where each $E_{ij}$ for $i \in I [1, m]$ and $j \in I [1, n]$ , is an $m \times n$ matrix with the element at position $(i, j)$ being one and the others being zero.

For two arbitrary integers $m$ and $n$ , $P (m, n)$ is the unitary matrix, i.e.

P (m, n) P^{T} (m, n) = P^{T} (m, n) P (m, n) = I_{mn} .

For two arbitrary integers $m$ and $n$ , $P (m, n) = P^{T} (n, m)$ .

Main results

This section consists of two parts. In the first part, the solvability of Problems 1 and 2 is discussed. In the second part, we give an iterative algorithm for solving the coupled linear matrix equations (1.5) over reflexive (anti-reflexive) matrices. In addition, sufficient conditions are established which guarantee the convergence of the proposed algorithm.

Solvability of the Problems 1 and 2

By some straightforward computations, we can prove the following lemmas.

Lemma 3.1. Problem 1 is solvable if and only if the system of matrix equations

(T_{ν} (X), T_{ν} (Z)) = (F_{ν}, F_{ν}), ν \in I [1, N],

(3.1)

are consistent, where $Z = (Z_{1}, Z_{2}, \dots, Z_{p})$ is defined such that $Z_{i} = R_{i} X_{i} Q_{i}$ in which the matrices $R_{i}, Q_{i} \in S O ℂ^{p_{i} \times p_{i}}$ are given for $i \in I [1, p]$ .

Proof. Without loss of generality, we may assume that $s_{1} = s_{2} = s_{3} = s_{4} = s$ . Suppose that the matrix group $X = (X_{1}, X_{2}, \dots, X_{p})$ is satisfied in Equation (3.1). Therefore,

{\begin{matrix} \sum_{i = 1}^{p} \sum_{μ = 1}^{s} A_{ν i μ} X_{i} B_{ν i μ} + C_{ν i μ} X_{i}^{T} D_{ν i μ} + M_{ν i μ} {\bar{X}}_{i} N_{ν i μ} + H_{ν i μ} X_{i}^{H} G_{ν i μ} = F_{ν} \\ \sum_{i = 1}^{p} \sum_{μ = 1}^{s} A_{ν i μ} R_{i} X_{i} Q_{i} B_{ν i μ} + C_{ν i μ} Q_{i}^{T} X_{i}^{t} R_{i}^{T} D_{ν i μ} + M_{ν i μ} {\bar{R}}_{i} {\bar{X}}_{i} {\bar{Q}}_{i} N_{ν i μ} + H_{ν i μ} Q_{i} X_{i}^{H} R_{i} G_{ν i μ} = F_{ν} . \end{matrix}

Now, we define the matrix group $\tilde{X} = ({\tilde{X}}_{1}, {\tilde{X}}_{2}, \dots, {\tilde{X}}_{p})$ such that

{\tilde{X}}_{i} = \frac{X_{i} + R_{i} X_{i} Q_{i}}{2}, i = 1, 2, \dots, p .

Evidently, $\tilde{X}$ is a group of reflexive matrices, i.e. ${\tilde{X}}_{i} \in ℂ_{r}^{p_{i} \times p_{i}} (R_{i}, Q_{i}),$ for $i = 1, 2, \dots, p .$ It is easy to verify that for $ν \in I [1, N]$

\begin{matrix} \sum_{i = 1}^{p} \sum_{μ = 1}^{s} A_{ν i μ} {\tilde{X}}_{i} B_{ν i μ} + C_{ν i μ} {\tilde{X}}_{i}^{T} D_{ν i μ} \\ + M_{ν i μ} {\bar{\tilde{X}}}_{i} N_{ν i μ} + H_{ν i μ} {\tilde{X}}_{i}^{H} G_{ν i μ} = F_{ν}, \end{matrix}

which shows that the matrix group $\tilde{X} = ({\tilde{X}}_{1}, {\tilde{X}}_{2}, \dots, {\tilde{X}}_{p})$ is the solution of Problem 1.

Conversely, let the matrix group $X = (X_{1}, X_{2}, \dots, X_{p})$ be the solution of Problem 1. That is

X_{i} \in ℂ_{r}^{p_{i} \times p_{i}} (R_{i}, Q_{i}), i = 1, 2, \dots, p,

and $X = (X_{1}, X_{2}, \dots, X_{p})$ is satisfied in Equation (1.5). Hence, the proof can be concluded immediately from the fact that $X_{i} = R_{i} X_{i} Q_{i}$ for $i = 1, 2, \dots, p .$ □

Analogous to the strategy employed in the proof of Lemma 3.1, we may establish the following lemma.

Lemma 3.2. Problem 2 is solvable if and only if the system of matrix equations

(T_{ν} (X), T_{ν} (Z)) = (F_{ν}, - F_{ν}), ν \in I [1, N] .

(3.2)

Assume that the matrices $ψ_{1}, ψ_{2}, ψ_{3}$ and $ψ_{4}$ are defined as follows:

ψ_{1} = (\begin{matrix} \sum_{μ = 1}^{s_{1}} B_{11 μ}^{T} \otimes A_{11 μ} & \dots & \sum_{μ = 1}^{s_{1}} B_{1 p μ}^{T} \otimes A_{1 p μ} \\ ⋮ & ⋮ \\ \sum_{μ = 1}^{s_{1}} B_{N 1 μ}^{T} \otimes A_{N 1 μ} & \dots & \sum_{μ = 1}^{s_{1}} B_{Np μ}^{T} \otimes A_{Np μ} \\ \sum_{μ = 1}^{s_{1}} B_{11 μ}^{T} Q_{1}^{T} \otimes A_{11 μ} R_{1} & \dots & \sum_{μ = 1}^{s_{1}} B_{1 p μ}^{T} Q_{p}^{T} \otimes A_{1 p μ} R_{p} \\ ⋮ & ⋮ \\ \sum_{μ = 1}^{s_{1}} B_{N 1 μ}^{T} Q_{1}^{T} \otimes A_{N 1 μ} R_{1} & \dots & \sum_{μ = 1}^{s_{1}} B_{Np μ}^{T} Q_{p}^{T} \otimes A_{Np μ} R_{p} \end{matrix}),

ψ_{2} = (\begin{matrix} \sum_{μ = 1}^{s_{2}} (D_{11 μ}^{T} \otimes C_{11 μ}) P (p_{1}, p_{1}) & \dots & \sum_{μ = 1}^{s_{2}} (D_{1 p μ}^{T} \otimes C_{1 p μ}) P (p_{p}, p_{p}) \\ ⋮ & ⋮ \\ \sum_{μ = 1}^{s_{2}} (D_{N 1 μ}^{T} \otimes C_{N 1 μ}) P (p_{1}, p_{1}) & \dots & \sum_{μ = 1}^{s_{2}} (D_{Np μ}^{T} \otimes C_{Np μ}) P (p_{p}, p_{p}) \\ \sum_{μ = 1}^{s_{2}} (D_{11 μ}^{T} R_{1} \otimes C_{11 μ} Q_{1}^{T}) P (p_{1}, p_{1}) & \dots & \sum_{μ = 1}^{s_{2}} (D_{1 p μ}^{T} R_{p} \otimes C_{1 p μ} Q_{p}^{T}) P (p_{p}, p_{p}) \\ ⋮ & ⋮ \\ \sum_{μ = 1}^{s_{2}} (D_{N 1 μ}^{T} R_{1} \otimes C_{N 1 μ} Q_{1}^{T}) P (p_{1}, p_{1}) & \dots & \sum_{μ = 1}^{s_{2}} (D_{Np μ}^{T} R_{p} \otimes C_{Np μ} Q_{p}^{T}) P (p_{p}, p_{p}) \end{matrix}),

ψ_{3} = (\begin{matrix} \sum_{μ = 1}^{s_{3}} N_{11 μ}^{T} \otimes M_{11 μ} & \dots & \sum_{μ = 1}^{s_{3}} N_{1 p μ}^{T} \otimes M_{1 p μ} \\ ⋮ & ⋮ \\ \sum_{μ = 1}^{s_{3}} N_{N 1 μ}^{T} \otimes M_{N 1 μ} & \dots & \sum_{μ = 1}^{s_{3}} N_{Np μ}^{T} \otimes M_{Np μ} \\ \sum_{μ = 1}^{s_{3}} N_{11 μ}^{T} Q_{1} \otimes M_{11 μ} {\bar{R}}_{1} & \dots & \sum_{μ = 1}^{s_{3}} N_{1 p μ}^{T} Q_{p} \otimes M_{1 p μ} {\bar{R}}_{p} \\ ⋮ & ⋮ \\ \sum_{μ = 1}^{s_{3}} N_{N 1 μ}^{T} Q_{1} \otimes M_{N 1 μ} {\bar{R}}_{1} & \dots & \sum_{μ = 1}^{s_{3}} N_{Np μ}^{T} Q_{p} \otimes M_{Np μ} {\bar{R}}_{p} \end{matrix}),

and

ψ_{4} = (\begin{matrix} \sum_{μ = 1}^{s_{4}} (G_{11 μ}^{T} \otimes H_{11 μ}) P (p_{1}, p_{1}) & \dots & \sum_{μ = 1}^{s_{4}} (G_{1 p μ}^{T} \otimes H_{1 p μ}) P (p_{p}, p_{p}) \\ ⋮ & ⋮ \\ \sum_{μ = 1}^{s_{4}} (G_{N 1 μ}^{T} \otimes H_{N 1 μ}) P (p_{1}, p_{1}) & \dots & \sum_{μ = 1}^{s_{4}} (G_{Np μ}^{T} \otimes H_{Np μ}) P (p_{p}, p_{p}) \\ \sum_{μ = 1}^{s_{4}} (G_{11 μ}^{tr} 1^{T} \otimes H_{11 μ} Q_{1}) P (p_{1}, p_{1}) & \dots & \sum_{μ = 1}^{s_{4}} (G_{1 p μ}^{tr} p^{T} \otimes H_{1 p μ} Q_{p}) P (p_{p}, p_{p}) \\ ⋮ & ⋮ \\ \sum_{μ = 1}^{s_{4}} (G_{N 1 μ}^{tr} 1^{T} \otimes H_{N 1 μ} Q_{1}) P (p_{1}, p_{1}) & \dots & \sum_{μ = 1}^{s_{4}} (G_{N p μ}^{tr} p^{T} \otimes H_{Np μ} Q_{p}) P (p_{p}, p_{p}) \end{matrix}) .

Let us assume that

Π = (ψ_{1} + ψ_{2}, ψ_{3} + ψ_{4}),

(3.3)

υ = (\begin{matrix} vec (X_{1}) \\ ⋮ \\ vec (X_{p}) \\ vec ({\bar{X}}_{1}) \\ ⋮ \\ vec ({\bar{X}}_{p}) \end{matrix}), Ω_{r} = (\begin{matrix} vec (F_{1}) \\ ⋮ \\ vec (F_{N}) \\ vec (F_{1}) \\ ⋮ \\ vec (F_{N}) \end{matrix}), and Ω_{a} = (\begin{matrix} vec (F_{1}) \\ ⋮ \\ vec (F_{N}) \\ - vec (F_{1}) \\ ⋮ \\ - vec (F_{N}) \end{matrix}) .

It is not difficult to see that Equations (3.1) and (3.2) are equivalent to the following to the following linear systems, respectively,

Π υ = Ω_{r},

and

Π υ = Ω_{a} .

Now, we can conclude the following theorems from Lemmas 3.1 and 3.2.

Theorem 3.3. Problem 1 has a unique solution if and only if $Rank (Π, Ω_{r}) = Rank (Π)$ and $Π$ has full column rank. The unique reflexive solution group of Problem (1) can be calculated as follows:

X_{i} = \frac{{\tilde{X}}_{i} + R_{i} {\tilde{X}}_{i} Q_{i}}{2}, i = 1, 2, \dots, p,

where $\tilde{X} = ({\tilde{X}}_{1}, \dots, {\tilde{X}}_{p}, {\bar{\tilde{X}}}_{1}, \dots, {\bar{\tilde{X}}}_{p})$ is derived such that

vec (\tilde{X}) = {(Π^{H} Π)}^{- 1} Π^{H} Ω_{r} .

Theorem 3.4. Problem 2 has a unique solution if and only if $Rank (Π, Ω_{a}) = Rank (Π)$ and $Π$ has full column rank. The unique reflexive solution group of Problem 2 can be calculated as follows:

X_{i} = \frac{{\tilde{X}}_{i} - R_{i} {\tilde{X}}_{i} Q_{i}}{2}, i = 1, 2, \dots, p,

where $\tilde{X} = ({\tilde{X}}_{1}, \dots, {\tilde{X}}_{p}, {\bar{\tilde{X}}}_{1}, \dots, {\bar{\tilde{X}}}_{p})$ is derived such that

vec (\tilde{X}) = {(Π^{H} Π)}^{- 1} Π^{H} Ω_{a} .

Proposed algorithm

In this section, by developing the idea of gradient based iterative method, we propose an iterative algorithm to compute the reflexive and anti-reflexive solutions of Equation (1.5).

Before presenting the algorithm, for $j = 1, 2, 3, 4$ , we define the linear operators $S_{j} (V)$ as follows

\begin{matrix} S_{j} : ℂ^{λ_{1} \times γ_{1}} \times \dots \times ℂ^{λ_{N} \times γ_{N}} \to ℂ^{p_{1} \times p_{1}} \times \dots \times ℂ^{p_{p} \times p_{p}} \\ V \to S_{j} (V) \end{matrix}

such that $V = (V_{1}, V_{2}, \dots, V_{N})$ , $S_{j} (V) = (S_{1 j} (V), S_{2 j} (V), \dots, S_{pj} (V))$ and

S_{i 1} (V) = \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{1}} A_{ν i μ}^{H} V_{ν} B_{ν i μ}^{H}, S_{i 2} (V) = \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{2}} {\bar{D}}_{ν i μ} V_{ν}^{T} {\bar{C}}_{ν i μ},

S_{i 3} (V) = \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{3}} {\bar{M}}_{ν i μ}^{H} {\bar{V}}_{ν} {\bar{N}}_{ν i μ}^{H}, S_{i 4} (V) = \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{4}} G_{ν i μ} V_{ν}^{H} H_{ν i μ},

for $i \in I [1, p]$ .

Algorithm 1 (Proposed algorithm for Problems 1 and 2).

Step 1. Input the matrices $A_{ν i μ} \in ℂ^{λ_{ν} \times p_{i}}, B_{ν i μ} \in ℂ^{p_{i} \times γ_{ν}}$ , $μ \in I [1, s_{1}]$ , $C_{ν i μ} \in ℂ^{λ_{ν} \times p_{i}}, D_{ν i μ} \in ℂ^{p_{i} \times γ_{ν}}$ , $μ \in I [1, s_{2}]$ , $M_{ν i μ} \in ℂ^{λ_{ν} \times p_{i}}, N_{ν i μ} \in ℂ^{p_{i} \times γ_{ν}}$ , $μ \in I [1, s_{3}]$ , $H_{ν i μ} \in ℂ^{λ_{ν} \times p_{i}}, G_{ν i μ} \in ℂ^{p_{i} \times γ_{ν}}$ , $μ \in I [1, s_{4}]$ , $F_{ν} \in ℂ^{λ_{ν} \times γ_{ν}}$ , $R_{i}, Q_{i} \in S O ℂ^{p_{i} \times p_{i}}$ , for $i \in I [1, p],$ and $ν \in I [1, N]$ .

Step 2. Set $κ = 0$ for computing the solution of Problem 1, and $κ = 1$ for Problem 2. Choose a tolerance $ε$ .

Step 3. If $κ = 0$ , select the initial reflexive group of matrices

X^{(1)} = (X_{1}^{(1)}, X_{2}^{(1)}, \dots, X_{p}^{(1)}) .

Otherwise select the initial anti-reflexive group of matrices

X^{(1)} = (X_{1}^{(1)}, X_{2}^{(1)}, \dots, X_{p}^{(1)}) .

Step 4. Set $t = 1$ . For $ν = 1, 2, 3, \dots, N,$ compute $R_{ν}^{(1)} = F_{ν} - T_{ν} (X^{(1)}) .$

Step 5. Determine the integer values $α_{ij}$ , for $i \in I [1, p]$ and $j = 1, 2, 3, 4$ such that if $S_{ij} (V) \equiv 0$ , for all $V = (V_{1}, V_{2}, \dots, V_{N})$ , set $α_{ij} = 0$ , where $V_{ν} \in ℂ^{λ_{ν} \times γ_{ν}}$ for $ν \in I [1, N]$ . Otherwise, set $α_{ij} = 1$ .

Step 6. For $i = 1, 2, \dots, p,$ Do:

If $α_{i 1} = 1$ , set

U_{i}^{(t + 1)} = X_{i}^{(t)} + \frac{ω}{2 N s_{1}} (S_{i 1} (R^{(t)}) + {(- 1)}^{κ} R_{i} S_{i 1} (R^{(t)}) Q_{i});

Otherwise $U_{i}^{(t + 1)} = 0 .$

If $α_{i 2} = 1$ , set

Y_{i}^{(t + 1)} = X_{i}^{(t)} + \frac{ω}{2 N s_{2}} (S_{i 2} (R^{(t)}) + {(- 1)}^{κ} R_{i} S_{i 2} (R^{(t)}) Q_{i});

Otherwise $Y_{i}^{(t + 1)} = 0 .$

If $α_{i 3} = 1$ , set

Z_{i}^{(t + 1)} = X_{i}^{(t)} + \frac{ω}{2 N s_{3}} (S_{i 3} (R^{(t)}) + {(- 1)}^{κ} R_{i} S_{i 3} (R^{(t)}) Q_{i});

Otherwise $Z_{i}^{(t + 1)} = 0 .$

If $α_{i 4} = 1$ , set

W_{i}^{(t + 1)} = X_{i}^{(t)} + \frac{ω}{2 N s_{4}} (S_{i 4} (R^{(t)}) + {(- 1)}^{κ} R_{i} S_{i 4} (R^{(t)}) Q_{i});

Otherwise $W_{i}^{(t + 1)} = 0 .$

Step 7. For $i = 1, 2, \dots, p,$ calculate $β_{i} : α_{i 1} + α_{i 2} + α_{i 3} + α_{i 4};$ Set

X_{i}^{(t + 1)} = \frac{1}{β_{i}} (U_{i}^{(t + 1)} + Y_{i}^{(t + 1)} + Z_{i}^{(t + 1)} + W_{i}^{(t + 1)}) .

Step 8. Compute $R^{(t + 1)} = (R_{1}^{(t + 1)}, R_{2}^{(t + 1)}, \dots, R_{N}^{(t + 1)})$ where

R_{ν}^{(t + 1)} = F_{ν} - T_{ν} (X^{(t + 1)}), ν = 1, 2, \dots, N .

Step 9. If $∥ R^{(t + 1)} ∥ < ε$ then Stop; otherwise, set $t = t + 1$ and go to Step 6.

Theorem 3.5. Suppose that the matrix equation (1.5) has unique reflexive solution group $X^{*} = (X_{1}^{*}, X_{2}^{*}, \dots, X_{p}^{*}) .$ If the parameter $ω$ satisfies the inequality

0 < ω < \frac{2}{Θ},

(3.4)

such that $Θ = \sum_{ν = 1}^{N} \sum_{i = 1}^{p} Θ_{i ν},$ where,

\begin{matrix} Θ_{1 ν} = \sum_{μ = 1}^{s_{1}} {‖ A_{ν i μ} ‖}^{2} {‖ B_{ν i μ} ‖}^{2}, Θ_{2 ν} = \sum_{μ = 1}^{s_{2}} {‖ C_{ν i μ} ‖}^{2} {‖ D_{ν i μ} ‖}^{2} \\ Θ_{3 ν} = \sum_{μ = 1}^{s_{3}} {‖ M_{ν i μ} ‖}^{2} {‖ N_{ν i μ} ‖}^{2}, Θ_{4 ν} = \sum_{μ = 1}^{s_{4}} {‖ H_{ν i μ} ‖}^{2} {‖ G_{ν i μ} ‖}^{2} . \end{matrix}

Then the iterative solution group, computed by Algorithm 1 (with $κ = 0$ ), $X^{(t)} = (X_{1}^{(t)}, X_{2}^{(t)}, \dots, X_{p}^{(t)}),$ converges for any initial reflexive matrix group

X^{(1)} = (X_{1}^{(1)}, X_{2}^{(1)}, \dots, X_{p}^{(1)}) .

Proof. Without loss of generality, we may assume that $α_{ij} = 1$ for $i \in I [1, p]$ and $j = 1, 2, 3, 4$ . For $i = 1, 2, \dots, p,$ we set

\begin{matrix} {\tilde{U}}_{i}^{(t)} = - \frac{U_{i}^{(t)} - X_{i}^{*}}{β_{i}}, {\tilde{Y}}_{i}^{(t)} = - \frac{Y_{i}^{(t)} - X_{i}^{*}}{β_{i}}, \\ {\tilde{Z}}_{i}^{(t)} = - \frac{Z_{i}^{(t)} - X_{i}^{*}}{β_{i}}, {\tilde{W}}_{i}^{(t)} = - \frac{W_{i}^{(t)} - X_{i}^{*}}{β_{i}} . \end{matrix}

It is not difficult to see that

\begin{matrix} {\tilde{L}}_{i}^{(t)} & : X_{i}^{*} - X_{i}^{(t)} = {\tilde{U}}_{i}^{(t)} + {\tilde{Y}}_{i}^{(t)} + {\tilde{Z}}_{i}^{(t)} + {\tilde{W}}_{i}^{(t)}, \\ i = 1, 2, \dots, p . \end{matrix}

For the matrix groups $U = (U_{1}, U_{2}, \dots, U_{p})$ , $Y = (Y_{1}, Y_{2}, \dots, Y_{p})$ , $Z = (Z_{1}, Z_{2}, \dots, Z_{p})$ and $W = (W_{1}, W_{2}, \dots, W_{p})$ , we define

T_{ν} (U, Y, Z, W) = T_{ν} (L), ν = 1, 2, \dots, N,

and the matrix group $L = (L_{1}, L_{2}, \dots, L_{p})$ is defined such that $L_{i} = (U_{i} + Y_{i} + Z_{i} + W_{i})$ for $i \in I [1, p]$ .

It is obvious that ${\tilde{L}}^{(t)} = X^{*} - X^{(t)}$ . Therefore,

\begin{matrix} T_{ν} ({\tilde{U}}^{(t)}, {\tilde{Y}}^{(t)}, {\tilde{Z}}^{(t)}, {\tilde{W}}^{(t)}) = T_{ν} (X^{*}) - T_{ν} (X^{(t)}) = R_{ν}^{(t)} . \end{matrix}

From Algorithm 1, with $κ = 0$ , it is known that $X^{(t - 1)} = (X_{1}^{(t - 1)}, X_{2}^{(t - 1)}, \dots, X_{p}^{(t - 1)})$ is a reflexive matrix group, i.e. $X_{i}^{(t - 1)} \in ℂ^{p_{i} \times p_{i}} (R_{i}, Q_{i})$ where the matrices $R_{i}, Q_{i} \in S O ℂ^{p_{i} \times p_{i}}$ are given and $i = 1, 2, \dots, p$ . For given arbitrary matrices $A, B$ , it is not difficult to see that $Re [tr (AB)] = Re [tr (BA)] = Re [tr (A^{H} B^{H})]$ . In addition, ${\tilde{L}}^{(t - 1)} = ({\tilde{L}}_{1}^{(t - 1)}, {\tilde{L}}_{2}^{(t - 1)}, \dots, {\tilde{L}}_{p}^{(t - 1)})$ is a group of reflexive matrices, i.e. ${\tilde{L}}_{i}^{(t - 1)} \in ℂ_{r}^{p_{i} \times p_{i}} (R_{i}, Q_{i})$ for $i \in I [1, p]$ . Therefore,

\begin{matrix} {‖ {\tilde{U}}^{(t)} ‖}^{2} \approx {‖ {\tilde{U}}^{(t - 1)} ‖}^{2} \\ - \frac{\bar{ω}}{4 N s_{1}} {\sum_{i = 1}^{p} Re [tr ({({\tilde{L}}_{i}^{(t - 1)})}^{H} (\sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{1}} A_{ν i μ}^{H} T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) B_{ν i μ}^{H} \\ + R_{i} A_{ν i μ}^{H} T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) B_{ν i μ}^{H} Q_{i}))]} \\ + \frac{{\bar{ω}}^{2}}{4 N^{2} s_{1}^{2}} {\sum_{i = 1}^{p} {‖ \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{1}} A_{ν i μ}^{H} T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) B_{ν i μ}^{H} + R_{i} A_{ν i μ}^{H} T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) B_{ν i μ}^{H} Q_{i} ‖}^{2}} \end{matrix}

\begin{matrix} \leq {‖ {\tilde{U}}^{(t - 1)} ‖}^{2} - \frac{\bar{ω}}{4 N s_{1}} \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{1}} \\ {Re [tr ((\sum_{i = 1}^{p} A_{ν i μ} {\tilde{L}}_{i}^{(t - 1)} B_{ν i μ}) {(T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}))}^{H})] \\ + Re [tr ((\sum_{i = 1}^{p} A_{ν i μ} R_{i} {\tilde{L}}_{i}^{(t - 1)} Q_{i} B_{ν i μ}) {(T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}))}^{H})]} \\ + \frac{2 {\bar{ω}}^{2}}{4 N s_{1}^{2}} {\sum_{i = 1}^{p} \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{1}} ‖ A_{ν i μ}^{H} T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) B_{ν i μ}^{H} \end{matrix}

\begin{matrix} {+ R_{i} A_{ν i μ}^{H} T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) B_{ν i μ}^{H} Q_{i} ‖}^{2}} \leq {‖ {\tilde{U}}^{(t - 1)} ‖}^{2} \\ - \frac{ω}{8 N s_{1}} {\sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{1}} Re [tr ((\sum_{i = 1}^{p} A_{ν i μ} {\tilde{L}}_{i}^{(t - 1)} B_{ν i μ}) {(T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}))}^{H})]} \\ + \frac{ω^{2}}{16 N s_{1}} {\sum_{i = 1}^{p} \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{1}} {‖ A_{ν i μ} ‖}^{2} {‖ B_{ν i μ} ‖}^{2} {‖ T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) ‖}^{2}} . \end{matrix}

In the above relations, we set $\bar{ω} = \frac{ω}{4}$ . Similarly, we can see that

\begin{matrix} {‖ {\tilde{Y}}^{(t)} ‖}^{2} \leq {‖ {\tilde{Y}}^{(t - 1)} ‖}^{2} \\ - \frac{ω}{8 N s_{2}} {\sum_{ν = 1}^{N} \sum_{μ = 1}^{S_{2}} Re [tr ((\sum_{i = 1}^{p} C_{ν i μ} {({\tilde{L}}_{i}^{(t - 1)})}^{T} D_{ν i μ}) {(T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}))}^{H})]} \\ + \frac{ω^{2}}{16 N s_{2}} {\sum_{i = 1}^{p} \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{2}} {‖ C_{ν i μ} ‖}^{2} {‖ D_{ν i μ} ‖}^{2} {‖ T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) ‖}^{2}}, \end{matrix}

\begin{matrix} {‖ {\tilde{Z}}^{(t)} ‖}^{2} \leq {‖ {\tilde{Z}}^{(t - 1)} ‖}^{2} \\ - \frac{ω}{8 N s_{3}} {\sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{3}} Re [tr ((\sum_{i = 1}^{p} M_{ν i μ} \bar{{\tilde{L}}_{i}^{(t - 1)}} N_{ν i μ}) {(T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}))}^{H})]} \\ + \frac{ω^{2}}{16 N s_{3}} {\sum_{i = 1}^{p} \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{3}} {‖ M_{ν i μ} ‖}^{2} {‖ N_{ν i μ} ‖}^{2} {‖ T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) ‖}^{2}}, \end{matrix}

and,

\begin{matrix} {‖ {\tilde{W}}^{(t)} ‖}^{2} \leq {‖ {\tilde{W}}^{(t - 1)} ‖}^{2} \\ - \frac{ω}{8 N s_{4}} {\sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{4}} Re [tr ((\sum_{i = 1}^{p} H_{ν i μ} {({\tilde{L}}_{i}^{(t - 1)})}^{H} G_{ν i μ}) {(T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}))}^{H})]} \\ + \frac{ω^{2}}{16 N s_{4}} {\sum_{i = 1}^{p} \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{4}} {‖ H_{ν i μ} ‖}^{2} {‖ G_{ν i μ} ‖}^{2} {‖ T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) ‖}^{2}} . \end{matrix}

Assume that

Λ^{(t)} = 16 N (s_{1} {‖ {\tilde{U}}^{(t)} ‖}^{2} + s_{2} {‖ {\tilde{Y}}^{(t)} ‖}^{2} + s_{3} {‖ {\tilde{Z}}^{(t)} ‖}^{2} + s_{4} {‖ {\tilde{W}}^{(t)} ‖}^{2}) .

By some straightforward computations, we get

\begin{array}{l} 0 \leq Λ^{(t)} \leq Λ^{(t - 1)} \\ - 2 ω {\sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{1}} Re [tr ((\sum_{i = 1}^{p} A_{ν i μ} {\tilde{L}}_{i}^{(t - 1)} B_{ν i μ}) {(T_{ν} {({\tilde{U}}^{(t - 1)})}^{T}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}))}^{H})] \\ + \sum_{μ = 1}^{s_{2}} Re [tr ((\sum_{i = 1}^{p} C_{ν i μ} {({\tilde{L}}_{i}^{(t - 1)})}^{T} D_{ν i μ}) {(T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}))}^{H})] \\ + \sum_{μ = 1}^{s_{3}} Re [tr ((\sum_{i = 1}^{p} M_{ν i μ} \bar{{\tilde{L}}_{i}^{(t - 1)}} N_{ν i μ}) {(T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}))}^{H})] \\ + \sum_{μ = 1}^{s_{4}} Re [tr ((\sum_{i = 1}^{p} H_{ν i μ} {({\tilde{L}}_{i}^{(t - 1)})}^{H} G_{ν i μ}) {(T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}))}^{H})]} \\ + ω^{2} {\sum_{i = 1}^{p} \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{1}} {‖ A_{ν i μ} ‖}^{2} {‖ B_{ν i μ} ‖}^{2} {‖ T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) ‖}^{2} \\ + \sum_{i = 1}^{p} \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{2}} {‖ C_{ν i μ} ‖}^{2} {‖ D_{ν i μ} ‖}^{2} {‖ T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) ‖}^{2} \\ + \sum_{i = 1}^{p} \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{3}} {‖ M_{ν i μ} ‖}^{2} {‖ N_{ν i μ} ‖}^{2} {‖ T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) ‖}^{2} \\ + \sum_{i = 1}^{p} \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{4}} {‖ H_{ν i μ} ‖}^{2} {‖ G_{ν i μ} ‖}^{2} {‖ T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) ‖}^{2}} \\ \leq Λ^{(t - 1)} - 2 ω \sum_{ν = 1}^{N} {‖ T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) ‖}^{2} \\ + ω^{2} {\sum_{i = 1}^{p} \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{1}} {‖ A_{ν i μ} ‖}^{2} {‖ B_{ν i μ} ‖}^{2} + \sum_{i = 1}^{q} \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{2}} {‖ C_{ν i μ} ‖}^{2} {‖ D_{ν i μ} ‖}^{2} \\ + \sum_{i = 1}^{p} \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{3}} {‖ M_{ν i μ} ‖}^{2} {‖ N_{ν i μ} ‖}^{2} + \sum_{i = 1}^{p} \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{4}} {‖ H_{ν i μ} ‖}^{2} {‖ G_{ν i μ} ‖}^{2}} \\ \times \sum_{ν = 1}^{N} {‖ T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) ‖}^{2} . \end{array}

As $ω$ satisfies in (3.4), by some straightforward computations, we have:

\begin{array}{l} 0 \leq Λ^{(t)} \leq Λ^{(t - 1)} - ω (2 - ω {\sum_{i = 1}^{p} \sum_{ν = 1}^{N} \sum_{μ = 1}^{s_{1}} {‖ A_{ν i μ} ‖}^{2} {‖ B_{ν i μ} ‖}^{2} + \sum_{μ = 1}^{s_{2}} {‖ C_{ν i μ} ‖}^{2} {‖ D_{ν i μ} ‖}^{2} \\ + \sum_{μ = 1}^{s_{3}} {‖ M_{ν i μ} ‖}^{2} {‖ N_{ν i μ} ‖}^{2} + \sum_{μ = 1}^{s_{4}} {‖ H_{ν i μ} ‖}^{2} {‖ G_{ν i μ} ‖}^{2}}) \times \sum_{ν = 1}^{N} {‖ T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) ‖}^{2} \\ = Λ^{(t - 1)} - ω (2 - ω Θ) \times \sum_{ν = 1}^{N} {‖ T_{ν} ({\tilde{U}}^{(t - 1)}, {\tilde{Y}}^{(t - 1)}, {\tilde{Z}}^{(t - 1)}, {\tilde{W}}^{(t - 1)}) ‖}^{2} \leq Λ^{(0)} - ω (2 - ω Θ) \times \sum_{ρ = 0}^{t - 1} \sum_{ν = 1}^{N} {‖ T_{ν} ({\tilde{U}}^{(ρ)}, {\tilde{Y}}^{(ρ)}, {\tilde{Z}}^{(ρ)}, {\tilde{W}}^{(ρ)}) ‖}^{2} . \end{array}

Therefore,

\begin{matrix} 0 \leq ω (2 - ω Θ) \times \sum_{ρ = 0}^{t - 1} \\ \sum_{ν = 1}^{N} {‖ T_{ν} ({\tilde{U}}^{(ρ)}, {\tilde{Y}}^{(ρ)}, {\tilde{Z}}^{(ρ)}, {\tilde{W}}^{(ρ)}) ‖}^{2} \leq Λ^{(0)}, \end{matrix}

which shows that

\begin{matrix} 0 \leq ω (2 - ω Θ) \times \sum_{ρ = 0}^{\infty} \\ \sum_{ν = 1}^{N} {‖ T_{ν} ({\tilde{U}}^{(ρ)}, {\tilde{Y}}^{(ρ)}, {\tilde{Z}}^{(ρ)}, {\tilde{W}}^{(ρ)}) ‖}^{2} \leq Λ^{(0)} < \infty . \end{matrix}

From the convergence theorem of series, we conclude that

lim_{t \to \infty} \sum_{ν = 1}^{N} {‖ T_{ν} ({\tilde{U}}^{(t)}, {\tilde{Y}}^{(t)}, {\tilde{Z}}^{(t)}, {\tilde{W}}^{(t)}) ‖}^{2} = 0 .

Hence,

lim_{t \to \infty} T_{ν} ({\tilde{U}}^{(t)}, {\tilde{Y}}^{(t)}, {\tilde{Z}}^{(t)}, {\tilde{W}}^{(t)}) = 0

or, equivalently,

lim_{t \to \infty} R_{ν}^{(t)} = 0, ν = 1, 2, \dots, N .

From Theorem 3.3, we can conclude the result immediately.□

With the same strategy as employed in the proof of Theorem 3.5, we may establish the following theorem.

Theorem 3.6. Suppose that the matrix equation (1.5) has unique anti-reflexive solution group $X^{*} = (X_{1}^{*}, X_{2}^{*}, \dots, X_{p}^{*}) .$ If the parameter $ω$ satisfies the inequality

0 < ω < \frac{2}{Θ},

(3.5)

such that $Θ = \sum_{ν = 1}^{N} \sum_{i = 1}^{p} Θ_{i ν},$ where

\begin{matrix} Θ_{1 ν} = \sum_{μ = 1}^{s_{1}} {‖ A_{ν i μ} ‖}^{2} {‖ B_{ν i μ} ‖}^{2}, Θ_{2 ν} = \sum_{μ = 1}^{s_{2}} {‖ C_{ν i μ} ‖}^{2} {‖ D_{ν i μ} ‖}^{2} & Θ_{3 ν} = \sum_{μ = 1}^{s_{3}} {‖ M_{ν i μ} ‖}^{2} {‖ N_{ν i μ} ‖}^{2}, Θ_{4 ν} = \sum_{μ = 1}^{s_{4}} {‖ H_{ν i μ} ‖}^{2} {‖ G_{ν i μ} ‖}^{2} . \end{matrix}

Then the iterative solution group, computed by Algorithm 1 (with $κ = 1$ ), $X^{(t)} = (X_{1}^{(t)}, X_{2}^{(t)}, \dots, X_{p}^{(t)}),$ converges for any initial anti-reflexive matrix group

X^{(1)} = (X_{1}^{(1)}, X_{2}^{(1)}, \dots, X_{p}^{(1)}) .

Numerical experiments

In this section, we present two numerical examples to show the effectiveness of the proposed algorithm. All of the numerical experiments presented in this section were computed in double precision with some Matlab codes on a Pentium 4 PC, with a 3.06 GHz CPU and 1.00 GB of RAM.

Example 4.1. We consider the following matrix equation

AXB + C X^{T} D + M \bar{X} N + H X^{H} G = F,

(4.1)

where

\begin{matrix} A = (\begin{matrix} 1 & 2 & 3 \\ 1 & - 1 & 1 \\ 1 & 1 & 1 \end{matrix}), B = (\begin{matrix} - 1 & 2 & 1 \\ 1 & - 3 & 1 \\ 1 & 2 & - 3 \end{matrix}), \\ C = (\begin{matrix} i & 1 & 1 \\ - 1 & 1 & 1 \\ i & 2 & 1 \end{matrix}), D = (\begin{matrix} 1 & - i & 1 \\ i & 1 & 1 \\ i & i & 1 \end{matrix}), \\ M = (\begin{matrix} 1 & i & 1 \\ 1 + i & 1 & 1 \\ - 1 & 1 & 1 \end{matrix}), N = (\begin{matrix} 1 & 2 & 2 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \end{matrix}), \\ H = (\begin{matrix} 1 & 1 & - 1 \\ 1 & 4 & 1 \\ - 2 & 0 & 2 \end{matrix}), G = (\begin{matrix} 1 & 0 & - 1 \\ 0 & i & 0 \\ - 1 & 0 & 1 \end{matrix}), \end{matrix}

and

F = (\begin{matrix} - 6 + 14 i & 1 - 5 i & 37 - 3 i \\ 7 + 7 i & - 3 - 4 i & 18 - 1 i \\ - 4 + 11 i & - 5 + 13 i & 10 - 1 i \end{matrix}) .

The exact solution of (4.1) is

X^{*} = (\begin{matrix} 2 + 2 i & 2 - 2 i & 1 \\ 1 & - 1 & 2 i \\ 2 - 2 i & 2 + 2 i & - 1 \end{matrix}),

which is $(R, Q)$ -reflexive, where

R = (\begin{matrix} 0 & 0 & - 1 \\ 0 & 1 & 0 \\ - 1 & 0 & 0 \end{matrix}), Q = (\begin{matrix} 0 & - 1 & 0 \\ - 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}) .

(4.2)

Here, we mention that $R, Q \in S O ℂ^{3 \times 3}$ .

We apply Algorithm 1 with $κ = 0$ to solve Problem 1 corresponding to system (4.1). The initial guess was taken to be the zero matrix $X_{1}^{(0)} = 0$ and the stopping criterion

δ_{k} = \frac{∥ R_{k} ∥_{F}}{∥ R_{0} ∥_{F}} < 10^{- 5}

was used, where $R_{k}$ is the residual of (4.1) at $k$ th iteration. For $ω = 0.0337,$ $0.250$ and $0.0200$ the method converges in 53, 73 and 92 iterations, respectively. For $ω = 0.0337$ , the computed solution is

X^{(53)} = (\begin{matrix} 2.00000 + 2.00002 i & 2.00002 - 1.99996 i & 1.00000 - 0.00007 i \\ 0.99999 - 0.00001 i & - 0.99999 + 0.00001 i & 0.00003 + 2.00000 i \\ 2.00002 - 1.99996 i & 2.00000 + 2.00002 i & - 1.00000 + 0.00007 i \end{matrix}),

which is in good agreement with the exact solution. In Figure 1 the convergence history of the method is displayed.

Figure 1.

Plot of $\log_{10} δ_{k}$ versus $k$ for the $(R, Q)$ -reflexive solution of (4.1).

We now consider (4.1) with the right-hand side

F = (\begin{matrix} 16 i & 3 + 19 i & - 5 - 31 i \\ 3 + 11 i & 1 + 12 i & - 3 i \\ 6 + 15 i & 13 + 9 i & 4 - 11 i \end{matrix}) .

The exact solution of the system is

X^{*} = (\begin{matrix} 0 & 0 & 1 + 2 i \\ 1 - 2 i & 1 - 2 i & 0 \\ 0 & 0 & 1 + 2 i \end{matrix}),

which is in $ℂ_{a}^{3 \times 3} (R, Q)$ where $R$ and $Q$ are defined by (4.2). All of the other assumptions are as before. Applying Algorithm 1 to the matrix equation with $κ = 1$ , the method converges in 82, 101 and 154 iterations for $ω = 0.0222,$ $0.0180$ and $0.0120$ , respectively. For $ω = 0.0222$ the computed solution is

X^{(82)} = (\begin{matrix} - 0.00001 + 0.00001 i & 0.00002 + 0.00003 i & 0.99999 + 2.00002 i \\ 0.99999 - 1.99995 i & 0.99999 - 1.99995 i & 0.00000 + 0.00000 i \\ - 0.00002 - 0.00003 i & 0.00001 - 0.00001 i & 0.99999 + 2.00002 i \end{matrix}) .

We observe that the method has provided a good approximate solution for the system. The convergence history is displayed in Figure (2).

Figure 2.

Plot of $\overset{10}{\log} δ_{k}$ versus $k$ for the $(R, Q)$ -anti-reflexive solution of (4.1).

Example 4.2. In this example, we consider the system of matrix equations

{\begin{matrix} X_{1} + A_{1} X_{1} B_{1} + M_{1} {\bar{X}}_{1} N_{1} + C_{2} X_{2}^{T} D_{2} + H_{2} X_{2}^{H} G_{2} = F_{1}, \\ X_{1} + {\tilde{A}}_{1} X_{1} {\tilde{B}}_{1} + {\tilde{M}}_{1} {\bar{X}}_{1} {\tilde{N}}_{1} + {\tilde{C}}_{2} X_{2}^{T} {\tilde{D}}_{2} + {\tilde{H}}_{2} X_{2}^{H} {\tilde{G}}_{2} = F_{2}, \end{matrix}

(4.3)

where

\begin{matrix} A_{1} = (\begin{matrix} 1 & 0 & 3 \\ - 1 & - 3 & 1 \\ 1 & 1 & 1 \end{matrix}), B_{1} = (\begin{matrix} - 1 & 2 & 1 \\ 1 & - 3 & 1 \\ 1 & 2 & - 3 \end{matrix}), M_{1} = (\begin{matrix} i & 1 & 1 \\ - 1 & 1 & 1 \\ i & 2 & 1 \end{matrix}) \\ N_{1} = (\begin{matrix} 1 & - i & 1 \\ i & 1 & 1 \\ 2 & i & 1 \end{matrix}), C_{2} = (\begin{matrix} 1 & i & 1 \\ 2 i & 1 & 0 \\ 0 & 1 & 1 \end{matrix}), D_{2} = (\begin{matrix} 1 & 2 & 2 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \end{matrix}), \\ H_{2} = (\begin{matrix} 1 & 1 & - 1 \\ 1 & 4 & 1 \\ - 2 & 0 & - 2 \end{matrix}), G_{2} = (\begin{matrix} 1 & 0 & - 1 \\ 0 & i & 0 \\ - 1 & 0 & 1 \end{matrix}), {\tilde{A}}_{1} = (\begin{matrix} 1 & 1 & 3 \\ 1 & - 1 & 1 \\ 1 & 1 & 1 \end{matrix}), \\ {\tilde{B}}_{1} = (\begin{matrix} - 1 & 2 & 1 \\ 1 & - 3 & 1 \\ 1 & 2 & - 3 \end{matrix}), {\tilde{M}}_{1} = (\begin{matrix} i & 0 & 1 \\ - 1 & 1 & - 2 \\ 0 & 2 & 1 \end{matrix}), {\tilde{N}}_{1} = (\begin{matrix} 1 & - i & 1 \\ 1 & 1 & 1 \\ i & i & 1 \end{matrix}) \\ {\tilde{C}}_{2} = (\begin{matrix} 1 & - i & 1 \\ i & 1 & 1 \\ - 1 & 1 & 1 \end{matrix}), {\tilde{D}}_{2} = (\begin{matrix} 1 & 2 & 2 \\ 1 & 0 & 0 \\ 0 & 1 & - 3 \end{matrix}), {\tilde{H}}_{2} = (\begin{matrix} 1 & 1 & - 1 \\ 1 & 4 & 1 \\ - 2 & 0 & 2 \end{matrix}), \\ {\tilde{G}}_{2} = (\begin{matrix} 1 & 0 & - 1 \\ - 1 & - i & 1 \\ - 1 & 2 & 1 \end{matrix}) . \end{matrix}

and

\begin{matrix} F_{1} = (\begin{matrix} 83 + 67 i & 88 + 66 i & 132 + 61 i \\ 58 + 183 i & 41 - 3 i & - 36 - 58 i \\ - 13 + 76 i & 54 - 90 i & 120 - 126 i \end{matrix}), \\ F_{2} = (\begin{matrix} 9 + 29 i & 74 - 67 i & - 12 + 95 i \\ 68 + 190 i & 99 - 36 i & - 118 - 215 i \\ 64 + 85 i & - 47 - 55 i & - 32 - 274 i \end{matrix}) . \end{matrix}

The exact solution of Equation (4.3) is the matrix group $(X_{1}^{*}, X_{2}^{*})$ where

\begin{matrix} X_{1}^{*} = (\begin{matrix} 2 + 2 i & 2 - 2 i & 1 \\ 1 & - 1 & 2 i \\ 2 - 2 i & 2 + 2 i & - 1 \end{matrix}), \\ X_{2}^{*} = (\begin{matrix} 16 + 16 i & 20 - 28 i & 12 + 4 i \\ 29 - 4 i & - 7 + 5 i & - 4 + 17 i \\ 10 - 19 i & 9 + 11 i & - 1 - 2 i \end{matrix}) . \end{matrix}

Here, $X_{1}^{*} \in ℂ_{r}^{3 \times 3} (R_{1}, Q_{1})$ where $R_{1} = R$ and $Q_{1} = Q$ are as in previous example and $X_{2}^{*} \in ℂ_{r}^{3 \times 3} (R_{2}, Q_{2})$ , where

\begin{matrix} R_{2} = \frac{1}{5} (\begin{matrix} 1 & 4 i & - 2 + 2 i \\ - 4 i & 1 & - 2 - 2 i \\ - 2 - 2 i & - 2 + 2 i & 3 \end{matrix}), \\ Q_{2} = \frac{1}{2} (\begin{matrix} 1 & - 1 - 1 i & - 1 i \\ - 1 + 1 i & 0 & - 1 - 1 i \\ 1 i & - 1 + 1 i & 1 \end{matrix}) . \end{matrix}

We apply Algorithm (1) with $κ = 0$ for solving system (4.3). The initial guess was taken to be the zero matrix group $(X_{1}^{(0)}, X_{2}^{(0)}) = (0, 0)$ and the stopping criterion

δ_{k} = max {\frac{∥ R_{i}^{(k)} ∥_{F}}{∥ R_{i}^{(0)} ∥_{F}} : i = 1, 2} < 10^{- 5}

was used, where $R_{k}^{(i)}$ is the residual of the $i$ th equation of (4.3) at iteration $k$ . For $ω = 0.036$ , $0.026$ and $0.016$ the method converges in 79, 110 and 181 iterations, respectively. For $ω = 0.036$ the computed solution is $(X_{1}^{(79)}, X_{2}^{(79)})$ , where

\begin{matrix} X_{1}^{(79)} = (\begin{matrix} 1.99988 + 2.00028 i & 1.99951 - 1.99997 i & 1.00011 - 0.00021 i \\ 0.99986 + 0.00016 i & - 0.99986 - 0.00016 i & 0.00009 + 1.99983 i \\ 1.99951 - 1.99997 i & 1.99988 + 2.00028 i & - 1.00011 + 0.00021 i \end{matrix}), \\ X_{2}^{(79)} = (\begin{matrix} 16.00000 + 16.00030 i & 19.99987 - 27.99978 i & 12.00068 + 3.99996 i \\ 28.99993 - 3.99999 i & - 6.99963 + 5.00038 i & - 4.00003 + 16.99934 i \\ 10.00041 - 19.00036 i & 8.99963 + 10.99996 i & - 1.00005 - 1.99997 i \end{matrix}) . \end{matrix}

As seen the computed solution is in good agreement with the exact solution. In Figure 3 the convergence history of the method is displayed.

Figure 3.

Plot of $\overset{10}{\log} δ_{k}$ versus $k$ for the $(R_{i}, Q_{i})$ -reflexive of (4.3).

We now change the right-hand side of (4.3) to

\begin{matrix} F_{1} = \frac{1}{10} (\begin{matrix} 148 + 24 i & 188 + 144 i & - 22 - 196 i \\ - 22 - 124 i & 142 - 44 i & 148 + 428 i \\ 77 + 104 i & 159 + 140 i & 59 - 100 i \end{matrix}), \\ F_{2} = \frac{1}{10} (\begin{matrix} 116 + 74 i & 142 + 232 i & - 58 - 150 i \\ 27 - 219 i & 374 + 406 i & - 249 + 423 i \\ - 12 + 68 i & 208 + 130 i & - 16 + 78 i \end{matrix}) . \end{matrix}

In this case, the exact solution of the system is $(X_{1}^{*}, X_{2}^{*})$ , where

\begin{matrix} X_{1}^{*} = (\begin{matrix} 0 & 0 & 1 + 2 i \\ 1 - 2 i & 1 - 2 i & 0 \\ 0 & 0 & 1 + 2 i \end{matrix}), \\ X_{2}^{*} = \frac{1}{10} (\begin{matrix} 18 - 20 i & 6 + 14 i & 6 i \\ 5 + 8 i & - 19 - 19 i & 4 - 15 i \\ 9 - 26 i & 11 - 25 i & 22 - 7 i \end{matrix}) . \end{matrix}

We have $X_{i}^{*} \in ℂ_{a}^{3 \times 3} (R_{i}, Q_{i})$ , for $i = 1, 2$ . All of the assumptions are as before. Algorithm 1 with $k = 1$ converges in 180, 223 and 336 iterations for $ω = 0.0222$ , $0.0180$ and $0.0120$ , respectively. For $w = 0.0222$ the computed solution by Algorithm (1) is $(X_{1}^{(180)}, X_{2}^{(180)})$ where

\begin{matrix} X_{1}^{(180)} = (\begin{matrix} 0.00003 + 0.00009 i & 0.00003 + 0.00010 i & 0.99998 + 1.99999 i \\ 0.99997 - 1.99999 i & 0.99997 - 1.99999 i & 0.00000 + 0.00000 i \\ - 0.00003 - 0.00010 i & - 0.00003 - 0.00009 i & 0.99998 + 1.99999 i \end{matrix}), \\ X_{2}^{(180)} = (\begin{matrix} 1.79996 - 2.00002 i & 0.60003 + 1.40002 i & - 0.00001 + 0.60000 i \\ 0.50000 + 0.79996 i & - 1.90004 - 1.89999 i & 0.40000 - 1.50003 i \\ 0.89999 - 2.60004 i & 1.10001 - 2.49999 i & 2.20000 - 0.70001 i \end{matrix}) . \end{matrix}

The convergence history of the method is shown in Figure 4.

Figure 4.

Plot of $\overset{10}{\log} δ_{k}$ versus $k$ for the $(R_{i}, Q_{i})$ -anti-reflexive of (4.3).

Based on our numerical results, analogous to Xie et al. (2009), it seems that that larger $ω$ may lead to a faster convergence rate. Using the matrix $Π$ (introduced by Equation (3.3)) and a similar approach to that examined by Wang and Liao (2012), we may obtain the optimum parameter $ω$ . Nevertheless, determining the optimum parameter depends on computing the largest and smallest eigenvalues of $Π$ which are expensive to calculate in general situations.

Conclusion

This paper has been devoted for finding the reflexive (anti-reflexive) solution group of a general class of complex coupled linear matrix equations. To this end, using a hierarchical identification principle, an iterative algorithm has been constructed. We have established that under certain conditions, the proposed algorithm converges to the reflexive (anti-reflexive) solution group of the mentioned coupled linear matrix equations for any initial reflexive (anti-reflexive) solution group. In order to illustrate the feasibility and effectiveness of the presented algorithm, some numerical experiments have been given.

Footnotes

Acknowledgements

The authors would like to thank three anonymous referees for their helpful comments and recommendations which helped to improve the quality of the paper.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

References

Beik

FPA

Salkuyeh

(2011) On the global Krylov subspace methods for solving general coupled matrix equation. Computers and Mathematics with Applications 62: 605–4613.

Bernstein

. Matrix Mathematics: Theory, Facts, and Formulas, 2nd edn. Princeton, NJ: Princeton University Press.

Bouhamidi

Jbilou

(2008) A note on the numerical approximate solutions for generalized Sylvester matrix equations with applications. Applied Mathematics and Computation 206: 687–694.

Chang

X-W

Wang

J-S

(1993) The symmetric solution of the matrix equations AX + YA = C, AXA^T + BYB^T = C and (A^TXA, B^TXB) = (C, D). Linear Algebra and its Applications 179: 171–189.

Chen

H-C

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

Chen

(2002) Special Matrices. Tsinghua University Press (in Chinese).

Dehghan

Hajarian

(2008) An iterative algorithm for solving a pair of matrix equation AYB = E, CYD = F over generalized centro-symmetric matrices. Computers and Mathematics with Applications 56: 3246–3260.

Dehghan

Hajarian

(2010) The general coupled matrix equations over generalized bisymmetric matrices. Linear Algebra and its Applications 432: 1531–1552.

Dehghan

Hajarian

(2011a) Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations. Applied Mathematical Modelling 35: 3285–3300.

10.

Dehghan

Hajarian

(2011b) Solving the generalized Sylvester matrix equation

\sum_{i = 1}^{p} A_{i} X B_{i} + \sum_{j = 1}^{q} C_{j} Y D_{j} = F

over reflexive and anti-reflexive matrices. International Journal of Control, Automation, and Systems 9: 118–124.

11.

Dehghan

Hajarian

(2012a) The generalised Sylvester matrix equations over the generalised bisymmetric and skew-symmetric matrices. International Journal of Systems Science 43: 1580–1590.

12.

Dehghan

Hajarian

(2012b) Two iterative algorithms for solving coupled matrix equations over reflexive and anti-reflexive matrices. Computational and Applied Mathematics 31: 353–371.

13.

Dehghan

Hajarian

(2012c) Construction of an iterative method for solving generalized coupled Sylvester matrix equations. Transactions of the Institute of Measurement and Control. DOI: 10.1177/0142331212465105.

14.

Ding

Chen

(2005a) Hierarchical identification of lifted state-space models for general dual-rate systems. IEEE Transactions on Circuits and Systems–I: Regular Papers 52: 1179–1187.

15.

Ding

Chen

(2005b) Hierarchical gradient-based identification of multivariable discrete-time systems. Automatica 41: 315–325.

16.

Ding

Chen

(2005c) Hierarchical least squares identification methods for multivariable systems. IEEE Transactions on Automatic Control 50: 397–402.

17.

Ding

Chen

(2005d) Gradient based iterative algorithms for solving a class of matrix equations. IEEE Transactions on Automatic Control 50: 1216–1221.

18.

Ding

Chen

(2005e) Iterative least squares solutions of coupled Sylvester matrix equations. Systems Control Letters 54: 95–107.

19.

Ding

Chen

(2006) On iterative solutions of general coupled matrix equations. SIAM Journal of Control Optimization 44: 2269–2284.

20.

Ding

Liu

(2011) Hierarchical least squares identification for linear SISO systems with dual-rate sampled-data. IEEE Transactions on Automatic Control 56: 2677–2683.

21.

Ding

Liu

Ding

(2008) Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle. Applied Mathematics and Computation 197: 41–50.

22.

Ding

Liu

Ding

(201) Iterative solutions to matrix equations of the form A_iXB_i = F_i. Computers and Mathematics with Applications 59: 3500–3507.

23.

Hajarian

Dehghan

(2011) The generalized centro-symmetric and least squares generalized centro-symmetric solutions of the matrix equation AYB + CY^TD = E. Mathematical Methods in the Applied Sciences 34: 1562–1579.

24.

Han

Xie

Ding

Liu

(2010) Hierarchical least squares based iterative identification for multivariable systems with moving average noises. Mathematical and Computer Modelling 51: 1213–1220.

25.

Huang

Ying

Gua

(2008) An iterative method for skew-symmetric solution and the optimal approximate solution of the matrix equation AXB = C. Journal of Computational and Applied Mathematics 212: 231–244.

26.

Jbilou

Riquet

(2006) Projection methods for large Lyapunov matrix equations. Linear Algebra and its Applications 415: 344–358.

27.

F-L

X-Y

Zhang

(2008) The generalized anti-reflexive solutions for a class of matrix equation (BX = C,XD = E). Computational and Applied Mathematics 27: 31–46.

28.

J-F

X-Y

Duan

X-F

Zhang

(2010a) Iterative method for mirror-symmetric solution of matrix equation AXB + CYD = E. Bulletin of the Iranian Mathematical Society 36: 35–55.

29.

Z-Y

Wang

Zhou

Duan

G-R

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

30.

Liang

You

Dai

(2007) An efficient algorithm for the generalized centro-symmetric solution of the matrix equation AXB = C. Numerical Algorithms 44: 173–184.

31.

Liu

Ding

Shi

(2012) Least squares estimation for a class of non-uniformly sampled systems based on the hierarchical identification principle. Circuits, Systems and Signal Processing 31: 1985–2000.

32.

Salkuyeh

Toutounian

(2006) New approaches for solving large Sylvester equations. Applied Mathematics and Computation 173: 9–18.

33.

Song

Chen

Zhao

(2011) Iterative solutions to coupled Sylvester-transpose matrix equations. Applied Mathematical Modelling 35: 4675–4683.

34.

Wang

(2011) A finite iterative algorithm for solving the generalized (P, Q)− reflexive solution of the linear systems of matrix equations. Mathematical and Computer Modelling 54: 2117–2131.

35.

Wang

Ding

Dong

(2012) Iterative parameter estimation for a class of multivariable systems based on the hierarchical identification principle and the gradient search. Circuits Systems and Signal Processing 31: 2167–2177.

36.

Wang

Liao

(2012) The optimal convergence factor of the gradient based iterative algorithm for linear matrix equations. Filomat 26: 607–613.

37.

Zhang

Duan

G-R

(2011a) Finite iterative solutions to coupled Sylvester-conjugate matrix equations. Applied Mathematical Modelling 35: 1065–1080.

38.

Duan

G-R

(2011b) Iterative algorithms for solving a class of complex conjugate and transpose matrix equations. Applied Mathematics and Computation 217: 8343–8353.

39.

Xie

Ding

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

40.

Xie

Liu

Yang

(2010) Gradient based and least squares based iterative algorithms for matrix equations AXB + CX^TD = F. Applied Mathematics and Computation 217: 2191–2199.

41.

Zhang

(2011) A note on the iterative solutions of general coupled matrix equation. Applied Mathematics and Computation 217 (2011) 9380–9386.

42.

Zhang

Ding

Liu

(2011) Hierarchical gradient based iterative parameter estimation algorithm for multivariable output error moving average systems. Computers and Mathematics with Applications 61: 672–682.

43.

Zhou

Duan

(2008) On the generalized Sylvester mapping and matrix equation. Systems Control Letters 57: 200–208.