矩阵方程A_1X_1B_1+A_2X_2B_2+…+A_lX_lB_l=C的中心对称解及其最佳逼近

The Centrosymmetric Solutions of Matrix Equation A_1X_1B_1+A_2X_2B_2+…+A_lX_lB_l=C and its Optimal Approximation

设矩阵X=(x_(ij))∈R^(n×n),如果x_(ij)=x_(n+1-i,n+1-j)(i,j=1,2,…,n),则称X是中心对称矩阵.该文构造了一种迭代法求矩阵方程A_1X_1B_1+A_2X_2B_2+…+A_lX_lB_l=C的中心对称解组(其中[X_1,X_2,…,X_l]是实矩阵组).当矩阵方程相容时,对任意初始的中心对称矩阵组[X_1^((0)),X_2^((0)),…,X_l^((0))],在没有舍入误差的情况下,经过有限步迭代,得到它的一个中心对称解组,并且,通过选择一种特殊的中心对称矩阵组,得到它的最小范数中心对称解组.另外,给定中心对称矩阵组[(?)_1,(?)_2,…,(?)_l],通过求矩阵方程A_1(?)_1B_1+A_2(?)_2B_2+…+A_l(?)_lB_l=(?)(其中(?)=C-A_1(?)_1B_1-A_2(?)_2B_2-…-A_l(?)_lB_l)的中心对称解组,得到它的最佳逼近中心对称解组.实例表明这种方法是有效的.

A matrix X = （xij） ∈ R^n×n is said to be centrosymmetric if xij = Xn＋1-i,n＋1-j （i,j = 1,2,…,n）. In this paper, an iterative method is constructed for finding the centrosymmetrie solutions of matrix equation A1X1B1 ＋ A2X2B2 ＋ … ＋ AlXlBl = C, where [X1,X2…,Xl] 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）,X^2（0）, , … , Xl^（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 centrosylnmetric matrix group . In addition, the optimal approximation centrosymmetric solution group to a given centrosymmetric matrix group [^-X1, ^-X2,…,^-Xl] in Frobenius norm can be obtained by finding the least norm centrosymmetric solution group of new matrix equation A1^～X1B1 ＋ A2^～X2B2 ＋…＋ Al^～XlBl = ^～C,where ^～C = C - A1^～X1B1 - A2^～X2B2 Al^～XlBl. Given numerical examples show that the iterative method is efficient.