An iterative method for symmetric solutions and optimal approximation solution of the system of matrix equations A₁XB₁ = C₁, A₂XB₂ = C₂

Abstract

The symmetric solutions of the system of matrix equations A₁XB₁ = C₁, A₂XB₂ = C₂ are too difficult to be obtained by applying matrices decomposition. In this paper, an iterative method is applied to solve this problem. With it, the solvability of this system of matrix equations can be determined automatically, when this system of matrix equations is consistent, its solution can be obtained within finite iterative steps, and its least-norm solution can be obtained by choosing a special kind of initial iterative matrix, furthermore, its optimal approximation solution to a given matrix can be derived by finding the least-norm symmetric solution of a new system of matrix equations $A_1\tilde{X}{B}_1={\tilde{C}}_1\text{,}{A}_2\tilde{X}{B}_2={\tilde{C}}_2$. Finally, numerical examples are given.

Introduction

Let R^m×n denote the set of all m × n real matrices, R^m = R^m×1, SR^n×n denote the set of all symmetric matrices in R^n×n. For a matrix A ∈ R^m×n, ∥A∥ represents its Frobenius norm, R(A) represents its column space, vec(·) represents the vec operator, i.e. $\mathrm{vec}(A)={\left(a_1^{\mathrm{T}}\text{,}{a}_2^{\mathrm{T}}\text{,}\dots \text{,}{a}_n^{\mathrm{T}}\right)}^{\mathrm{T}}$ for the matrix A = (a₁, a₂, … , a_n) ∈ R^m×n, a_i ∈ R^m, i = 1, 2, … , n. A ⊗ B stands for the Kronecker product of matrices A and B.

In this paper, we consider the following problems.

Problem I

Given $A_1\in R^{m_1\times n}$, $B_1\in R^{n\times p_1}$, $C_1\in R^{m_1\times p_1}$, $A_2\in R^{m_2\times n}$, $B_2\in R^{n\times p_2}$, $C_2\in R^{m_2\times p_2}$, find X ∈ SR^n×n, such that

$$\left\{\begin{array}{c}{A}_1{\mathit{XB}}_1=C_1\text{,}\hfill \\ A_2{\mathit{XB}}_2=C_2.\hfill \end{array}\right.$$

Problem II

When Problem I is consistent, let S_E denote the set of solutions of Problem I, for given X₀ ∈ R^n×n, find $\hat{X}\in S_E$, such that

$$\Vert \hat{X}-X_0\Vert =\underset{X\in S_E}{\min }\Vert X-X_0\Vert \text{.}$$

In fact, Problem II is to find the optimal approximation solution to the given matrix X₀.

Research on solving systems of linear matrix equations has been actively ongoing for past years. For e.g., Mitra [1], [2] has provided conditions for the existence of a solution and a representation of the general common solution to the matrix equations AX = C, XB = D and the matrix equations A₁XB₁ = C₁, A₂XB₂ = C₂. Also, Navarra et al. [3] studied a representation of the general common solution to the matrix equations A₁XB₁ = C₁, A₂XB₂ = C₂, Bhimasankaram [4] considered the linear matrix equations AX = B, CX = D and EXF = G, van der Woude [5] obtained the existence of a common solution X to the matrix equations A_iXB_j = C_ij, (i, j) ∈ Γ. Till now, the problem to obtain symmetric solutions of the system of matrix equations A₁XB₁ = C₁, A₂XB₂ = C₂ has not been solved.

In this paper, an iterative method is constructed to solve the system of matrix equations A₁XB₁ = C₁, A₂XB₂ = C₂ over symmetric X. With it, the solvability of the system of matrix equations can be determined automatically, when the system of matrix equations is consistent, its symmetric solution can be obtained within finite iterative steps, and its least-norm solution can be obtained by choosing a suitable initial iterative matrix, furthermore, its optimal approximation solution to a given matrix can be derived by finding the least-norm symmetric solution of a new system of matrix equation $A_1\tilde{X}{B}_1={\tilde{C}}_1\text{,}{A}_2\tilde{X}{B}_2={\tilde{C}}_2$.