具最小二乘谱约束的结构矩阵逼近问题及其扰动分析

数学物理学报 ›› 2013, Vol. 33 ›› Issue (1): 37-45.

具最小二乘谱约束的结构矩阵逼近问题及其扰动分析

谢冬秀^1,2, 张忠志³

1.北京信息科技大学理学院北京 100192|2.湖南大学数学与计量经济学院长沙 410082|3.东莞理工学院数学系广东东莞 523808

收稿日期:2011-11-09 修回日期:2012-11-13 出版日期:2013-02-25 发布日期:2013-02-25
基金资助:
北京市自然科学基金(1122015)和北京市属高等学校人才强教深化计划项目(PHR201006116)资助资助

Structured Matrix Nearness Problem with Least Square Spectra Constraint and Its Perturbation Analysis

XIE Dong-Xiu^1,2, ZHANG Zhong-Zhi³

1.School of Science, Beijing Information Science and Technology University, Beijing 100192;
2.College of Mathematics and Econometrics, Hunan University, Changsha 410082;
3.Department of Mathematics, Dongguan University of Technology, Guangdong Dongguan 523808

Received:2011-11-09 Revised:2012-11-13 Online:2013-02-25 Published:2013-02-25
Supported by:
北京市自然科学基金(1122015)和北京市属高等学校人才强教深化计划项目(PHR201006116)资助资助

摘要/Abstract

摘要：

从两个方面讨论具有最小二乘谱约束的对称斜哈密尔顿矩阵的逼近问题: (I) 研究使AX − X∧ 的Frobenius范数最小的n阶实对称斜哈密尔顿矩阵A的集合L, 其中X, ∧分别是特征向量和特征值矩阵, (II) 求 Â ∈ L使得C − Â = min _A∈L ||C −A||, 这里 || ·|| 是Frobenius范数. 给出了L的元素的一般表达式和Â的显示表达式, 分析了该最佳逼近矩阵Â的扰动理论, 并给出了数值实验.

关键词: 最佳逼近, 最小二乘问题, 扰动理论

Abstract:

A nearness matrix problem is considered with two constraints—least square spectra constraint, symmetric and skew-Hamiltonian structure. It discusses two problems: (I) the set L of symmetric and skew-Hamiltonian real n × n matrices A to minimize the Frobenius norm of AX − X∧, where X, ∧ are eigenvector and eigenvalue matrices, respectively, and (II) find Â ∈ L such that C − Â = min _A∈L ||C −A||, where || ·|| is the Frobenius norm. A general form of elements in L is given and an explicit expression of the minimizer Â is derived. Perturbation theory of the nearest matrix is analyzed. A numerical example is reported.

Key words: Best approximation, Least squares problem, Perturbation theory

中图分类号:

41A50

谢冬秀, 张忠志. 具最小二乘谱约束的结构矩阵逼近问题及其扰动分析[J]. 数学物理学报, 2013, 33(1): 37-45.

XIE Dong-Xiu, ZHANG Zhong-Zhi. Structured Matrix Nearness Problem with Least Square Spectra Constraint and Its Perturbation Analysis[J]. Acta mathematica scientia,Series A, 2013, 33(1): 37-45.

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