矩阵方程AX=B的循环解及其最佳逼近

Circulant Solution of Matrix Equation AX=B and its Optimal Approximation

文章首先考虑了如下问题:给定矩阵A,B∈Cn×m,求循环矩阵X∈CIRn×n,使得minX‖AX-B‖。给出了问题具有循环矩阵解的条件和解的一般表达式,若用SE表示上述问题解的集合,文章还考虑了最佳逼近问题:给定X*∈CIRn×n,求X∈SE,使得minX∈SE‖X-X*‖=‖X-X*‖,其中‖·‖表示矩阵的Frobenius范数,证明了问题存在唯一解,给出了其唯一解的一般表达式。

In this paper, we first coneider the problelm as follow：Find a circulant matrix X∈CIRn×n such that for given matrics A,X∈Cn×m we have min ｜｜ AX - B ｜｜The existence theorems are obtained,and a general representation of such a matrix is presented. We denote the set of such ma- trices by SE. Then the matrix approximation problem is discussed. That is： Find a matrixX∈ SE such that for a given X∈ CIRn×n X∈CIRn×nwe have minX∈SE｜｜X-X＊｜｜=｜｜X-X＊｜｜ Where ｜｜·｜｜ is the Frobenius norm of matrics. We show that the approximation matrix is unique and provide an expression for thisapproximation matrix.