矩阵方程AX+XB=C的对称解及其最佳逼近

The Symmetric Solution of Matrix Equation and Its Optimal Approximation

提出一种求解线性矩阵方程AX+XB=C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对称解,在方程不相容时得到方程的最小二乘对称解.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对称解.若取特殊的初始矩阵,则得到问题的极小范数对称解,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 symmetric solution of the linear matrix equation AX ＋ XB = C is put forward in this paper. This iterative method can judgeautomatically the information of solutions. When the equation is consistent, it con-verges a symmetric solution of the equation. When the equation is inconsistent, it converges the least-squares symmetric solution of the equation. More over, for any initial matrix, a symmetric solution can be obtained within finite iteration steps in theabsence of roundoff errors. If a special kind of initial matrix is chosen, the syrnmetricsolution with least norm can be obtained, which wonderfully handle the problem of solving its optimal approximation solution for a given matrix.