| Least-Squares Solution of Inverse Problem for Hermitian Anti-reflexive Matrices and Its Appoximation |
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| 作者姓名： | Zhen Yun PENG Yuan Bei DENG Jin Wang LIU |
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| 作者单位： | [1]Department of Mathematics, Hunan University of Science and Technology, Xiangtan 411201, P. R. China [2]Department of Mathematics, Central South University, Changsha 410083, P. R. China [3]Institute of Computational Mathematics, Chinese Academy of Sciences, Beijing 100080, P. R. China [4]Technology, Department of Mathematics, Hunan University of Science and Xiangtan 411201, P, R, China |
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| 基金项目： | This work is supported by China Postdoctoral Science Foundation （Grant No. 2004035645）. The authors are very grateful to the referee for valuable comments. |
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| 摘 要： | In this paper, we first consider the least-squares solution of the matrix inverse problem as follows： Find a hermitian anti-reflexive matrix corresponding to a given generalized reflection matrix J such that for given matrices X, B we have minA ｜｜AX - B｜｜. The existence theorems are obtained, and a general representation of such a matrix is presented. We denote the set of such matrices by SE. Then the matrix nearness problem for the matrix inverse problem is discussed. That is： Given an arbitrary A^＊, find a matrix A E SE which is nearest to A^＊ in Frobenius norm. We show that the nearest matrix is unique and provide an expression for this nearest matrix.
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| 关 键 词： | 最小平方解 逆矩阵 反射矩阵 存在性 |
| 收稿时间： | 2003-08-20 |
| 修稿时间： | 2003-08-202004-11-11 |
Least–Squares Solution of Inverse Problem for Hermitian Anti–reflexive Matrices and Its Appoximation |
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| Zhen Yun PENG Yuan Bei DENG Jin Wang LIU.Least-Squares Solution of Inverse Problem for Hermitian Anti-reflexive Matrices and Its Appoximation[J].Acta Mathematica Sinica,2006,22(2):477-484. |
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| Authors: | Zhen Yun Peng Yuan Bei Deng Jin Wang Liu |
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| Institution: | (1) Department of Mathematics, Hunan University of Science and Technology, Xiangtan 411201, P. R. China;(2) Department of Mathematics, Central South University, Changsha 410083, P. R. China;(3) Institute of Computational Mathematics, Chinese Academy of Sciences, Beijing 100080, P. R. China;(4) Department of Mathematics, Hunan University of Science and Technology, Xiangtan 411201, P. R. China |
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| Abstract: | In this paper, we first consider the least-squares solution of the matrix inverse problem as follows: Find a hermitian anti–reflexive matrix corresponding to a given generalized reflection matrix J such that for given matrices X,B we have min A ‖AX ?B‖. The existence theorems are obtained, and a general representation of such a matrix is presented. We denote the set of such matrices by S E . Then the matrix nearness problem for the matrix inverse problem is discussed. That is: Given an arbitrary A*, find a matrix  ∈ S E which is nearest to A* in Frobenius norm. We show that the nearest matrix is unique and provide an expression for this nearest matrix. |
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| Keywords: | hermitian reflexive matrix hermitian anti-reflexive matrix matrix norm nearest matrix |
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