| 矩阵方程AX=B的循环解及其最佳逼近 |
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| 引用本文： | 张湘林,李云翔. 矩阵方程AX=B的循环解及其最佳逼近[J]. 衡阳师专学报, 2011, 0(6): 29-32 |
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| 作者姓名： | 张湘林 李云翔 |
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| 作者单位： | 湖南城市学院数学与计算科学系,湖南益阳413000 |
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| 基金项目： | 基金项目：湖南省教育厅资助项目（10C0501） |
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| 摘 要： | 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min｜｜AX—B｜｜。给X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XESE数，证明了问题存在唯一解，给出了其唯一解的一般表达式。
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| 关 键 词： | 矩阵方程 循环矩阵 矩阵范数 最佳逼近矩阵 |
Circulant Solution of Matrix Equation AX=B and its Optimal Approximation |
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| ZHANG Xiang-lin,LI Yun-xiang. Circulant Solution of Matrix Equation AX=B and its Optimal Approximation[J]. Journal of Hengyang Normal University, 2011, 0(6): 29-32 |
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| Authors: | ZHANG Xiang-lin LI Yun-xiang |
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| Affiliation: | (Department of Mathematics and Computing Science, Hunan City University, Yiyang Hunan 413000, China) |
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| Abstract: | 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. |
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| Keywords: | matrix equation circulant matrix matrix norm the approximation matrix |
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