线性流形上的广义反射矩阵反问题

数学物理学报 ›› 2009, Vol. 29 ›› Issue (6): 1547-1560.

线性流形上的广义反射矩阵反问题

1.南京航空航天大学理学院南京 210016; 2. 江苏科技大学数理学院江苏镇江 212003

收稿日期:2007-12-09 修回日期:2008-11-07 出版日期:2009-12-25 发布日期:2009-12-25
基金资助:
国家自然科学基金(10271055) 资助

Inverse Problems for Generalized Reflexive Matrices on a Linear Manifold

1.Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing |210016;
2.School of Mathematics and Physics, Jiangsu University of Science and Technology, Jiangsu Zhenjiang 212003

Received:2007-12-09 Revised:2008-11-07 Online:2009-12-25 Published:2009-12-25
Supported by:
国家自然科学基金(10271055) 资助

摘要/Abstract

摘要：

设 R ∈C^m×m 及 S ∈C^n×n 是非平凡Hermitian酉矩阵, 即 R^H=R=R^-1 ≠ ± I_m , S^H=S=S^-1 ≠ ± I_n.若矩阵 A ∈C^m×n 满足 RAS=A, 则称矩阵 A 为广义反射矩阵. 该文考虑线性流形上的广义反射矩阵反问题及相应的最佳逼近问题. 给出了反问题解的一般表示, 得到了线性流形上矩阵方程 AX₂=Z₂, Y₂^HA=W₂^H具有广义反射矩阵解的充分必要条件, 导出了最佳逼近问题唯一解的显式表示.

关键词: 反问题, 最佳逼近, 广义反射矩阵

Abstract:

Let R ∈C^m×m and S ∈C^n×n be nontrivial unitary involutions, i.e., R^H=R=R^-1≠ ± I_m and S^H=S=S^-1≠ ± I_n. A ∈C^m×n is said to be a generalized reflexive matrixif RAS=A. This paper is concerned with the inverse problem for generalized reflexive matrices on a linear manifold and the optimal approximation to a given matrix. The general expression of the solutions of the problem is presented. Sufficient and necessary conditions for equations AX₂=Z₂, Y₂^HA=W₂^H having a common generalized reflexive matrix solution on the linear manifold are derived. The expression of the solution for relevant optimal approximation problem is given.

Key words: Inverse problem, Optimal approximation, Generalized reflexive matrix

中图分类号:

15A24

袁永新, 戴华. 线性流形上的广义反射矩阵反问题[J]. 数学物理学报, 2009, 29(6): 1547-1560.

YUAN Yong-Xin, DAI Hua. Inverse Problems for Generalized Reflexive Matrices on a Linear Manifold[J]. Acta mathematica scientia,Series A, 2009, 29(6): 1547-1560.

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