矩阵方程AX+XB=C的双对称解及其最佳逼近

The Bisymmetric Solution of Matrix Equation AX+XB=C and Its Optimal Approximation

提出一种求解线性矩阵方程AX+XB=C双对称解的迭代法.该算法能够自动地判断解的情况,并在方程相容时得到方程的双对称解,在方程不相容时得到方程的最小二乘双对称解.对任意的初始矩阵,在没有舍入误差的情况下,经过有限步迭代得到问题的一个双对称解.若取特殊的初始矩阵,则可以得到问题的极小范数双对称解,从而巧妙地解决了对给定矩阵求最佳逼近解的问题.

An iterative method to find the bisymmetric solution of the linear matrix equation AX＋XB=C is put forward in this paper.This iterative method can judge automatically the information of solutions.When the equation is consistent,it converges a bisymmetric solution of the equation.When the equation is inconsistent,It converges the least-squares bisymmetric solution of the equation.For any initial matrix,a bisymmetric solution can be obtained within finite iteration steps in the absence of roundoff errors.If a special kind of initial matrix is chosen,the bisymmetric solution with least norm can be obtained,and wonderfully handle the problem of solving its optimal approximation solution for a given matrix.