矩阵方程AX=B的双反对称最佳逼近解

The Anti-bisymmetric Matrices Optimal Approximation Solution of Matrix Equation AX=B

本文主要讨论下面两个问题并得到相关结果:问题Ⅰ:给定A∈R^(k×n),B∈R^(k×n),求X∈BASR^(n×n),使得AX=B.问题Ⅱ:给定X~*∈R^(n×n),求■使得‖■-X~*‖=(?)‖X-X~*‖,其中S_E是问题Ⅰ的解集合,‖·‖是Frobenius范数.通过对上述问题的讨论给出了问题Ⅰ解存在的充分必要条件和其解的一般表达式同时给出了问题Ⅱ的解,算法,和数值例子.

This paper is mainly concerned with solving the following two problems,ProblemⅠ：Given k×n real matrices A and B,find X∈BASR^（n×n） such that AX=B.ProblemⅡ：Given an n×n real matrix X^＊,find an n×n matrix X such that‖X-X^＊‖=minX∈SE‖X-X^＊‖,whereⅡ·Ⅱis a Frobenius norm,and S_E is the solution set of ProblemⅠ. The necessary and sufficient conditions for the existence and expressions of the generalsolutions of ProblemⅠare given.The explicit solution,a numerical algorithm and anumerical example to ProblemⅡare provided.