关键词: 迭代法, 矩阵方程, 中心对称解组, 最小范数解组, 最佳逼近解组.

Abstract: A matrix X=(x_ij)∈R^n×n is said to be centrosymmetric if x_ij=x_{n+1-i, n+1-j}(i, j=1,2, …, n). In this paper, an iterative method is constructed for finding the centrosymmetric solutions of matrix equation A₁X₁B₁+A₂X₂B₂+…+A_lX_lB_l=C, where [X₁, X₂,…, X_l] is a real matrix group. By this iterative method, the solvability of the matrix equation can be judged automatically. When the matrix equation is consistent, for any initial centrosymmetric matrix group
[X₁⁽⁰⁾, X₂⁽⁰⁾,…, X_l⁽⁰⁾, a centrosymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors, and the least norm centrosymmetric solution group can be obtained by choosing a special kind of initial centrosymmetric matrix group . In addition, the optimal approximation centrosymmetric solution group to a given centrosymmetric matrix group
[X₁, X₂, …, X_l] in Frobenius norm can be obtained by finding the least norm centrosymmetric solution group of new matrix equation A1X₁B₁+A₂X₂B₂+…+A_lX_lB_l=C, where C=C-A₁X₁B₁-A₂X₂B₂-…-A_lX_lB_l. Given numerical examples show that the iterative method is efficient.

Key words: Iterative method, Matrix equation, Centrosymmetric solution group, Least-norm solution group, Optimal approximation solution.