求矩阵方程组A_1XB_1=C_1,A_2XB_2=C_2最小二乘对称解及其最佳逼近的迭代法

An Iterative Method for the Least Squares Symmetric Solution of the Matrix Equations A_1XB_1=C_1,A_2XB_2=C_2 and Its Optimal Approximation

该文提出了梯度矩阵（△↓F（X））的概念，构造了一种迭代法求最小二乘问题min‖（A1XB1，A2XB2）-（C1，C2）‖的对称解.通过这种方法，给定初始对称矩阵X1，在没有舍入误差的情况下，经过有限步迭代，找到它的一个对称解.并且，通过选择一种特殊的初始对称矩阵，得到它的最小范数对称解X＊.另外，给定对称矩阵X0，通过求解最小二乘问题min‖（A1X^-B1，A2X^-B2）-（C^-1，C^-2）‖（其中C^-1=C1-A1X0B1，C^-2=C2-A2X0B2），得到它的最佳逼近对称解.

In this paper, the concept of gradient matrix （ △↓ F（X） ） is presented, and an algorithm is constructed to solve the symmetric solution of the minimum Frobenius norm residual problem： min ｜｜ （ A1XB1, A2XB2 ） - （ C1, C2 ）｜｜. By this algorithm ,for any initial symmetric matrix X1 , a solution X＊ can be obtained within finite iteration steps in the absence of roundoff errors, and the solution X＊ with least norm can be obtained by choosing a special kind of initial symmetric matrix. In addition, the unique optimal approximation solution X^^ for a given matrix X0 in the Frobenius norm can be obtained by finding the least noml symmetric solution X^-＊ of the new minimum residual problem： rain ｜｜（A1X^-B1, A2X^-B2） - （C^-1,C^-2）｜｜, where C^-1 = C1 - A1X0B1,C^-2 = C2 - A2X0B2 .