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.

参考文献

[1] Chen H C. Generalized reflexive matrices: special properties and applications. SIAM J Matrix Anal Appl, 1998, f 19: 140--153

[2] Chen H C. The SAS domain decomposition method for structural analysis. CSRD Tech Report 754, Center for
Supercomputing Research and Development. Urbana, IL: University of Illinois, 1988

[3] Chen W, Wang X, Zhong T. The structure of weighting coefficient matrices of harmonic differential quadrature and its application. Comm Numer Methods Eng, 1996, 12: 455--460

[4] Datta L, Morgera S. On the reducibility of centrosymmetric matrices-applications in engineering problems.
Circuits Systems Signal Process, 1989, 8: 71--96

[5] Delmas J. On adaptive EVD asymtotic distribution of centro-symmetric covariance matrices. IEEE Trans Signal
Process, 1999, 47: 1402--1406

[6] Weaver J R. Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, eigenvectors. Amer Math Monthly, 1985, 92: 711--717

[7] Andrew A L. Eigenvectors of certain matrices. Linear Algebra Appl, 1973, 7: 151--162

[8] Andrew A L. Solution of equations involving centrosymmetric matrices. Technometrics, 1973, 15: 405--407

[9] Cantoni A, Butler P. Eigenvalues and eigenvectors of symmetric centrosymmetric matrices. Linear Algebra Appl, 1976, 13: 275--288

[10] Pye W C, Boullino T L, Atchison T A. The pseudoinverse of a centrosymmetric matrix. Linear Algebra Appl, 1973, 6: 201--204

[11] Tao D, Yasuda M. A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices. SIAM J Matrix Anal, 2002, 23: 885--895

[12] Yasuda M. Spectral characterizations for Hermitian centrosymmetric K-matrices and Hermitian skew-centrosymmetric K-matrices. SIAM J Matrix Anal, 2003, 25: 601--605

[13] Chu M T. Inverse eigenvalue problems. SIAM Rev, 1998, 40: 1--39

[14] Chu M T, Golub G H. Structured inverse eigenvalue problems. Acta Numer, 2002, 11: 1--71

[15] Chu M T, Golub G H. Inverse Eigenvalue Problems, Theory, Algorithms and Applications. Oxford: Oxford University Press, 2005

[16] Xu S F. An Introduction to Inverse Algebraic Eigenvalue Problems. Beijing: Peking University Press, 1998

[17] 周树荃, 戴华. 代数特征值反问题. 郑州: 河南科技出版社, 1991

[18] Bai Z J, Chan R H. Inverse eigenproblem for centrosymmetric and centroskew matrices and their approximation. Theoretical Computer Science, 2004, 315: 309--318

[19] Zhou F Z, Zhang L, Hu X Y. Least-squares solution for inverse problems of centrosymmetric matrices. Comput Math Appl, 2003, 45: 1581--1589

[20] Li F L, Hu X Y, Zhang L. Left and right inverse eigenpairs problem of skew-centrosymmetric matrices. Appl
Math Comput, 2006, 177: 105--110

[21] Zhou F Z. The solvability conditions for the inverse eigenvalue problems of reflexive matrices. J Comput Appl
Math, 2006, 188: 180--189

[22] Peng Z Y, Hu X Y. The reflexive and anti-reflexive solutions of the matrix equation AX=B. Linear Algebra Appl, 2003, 375: 147--155

[23] 袁永新, 戴华. 线性流形上的广义中心对称矩阵反问题. 计算数学, 2005, 27: 383--394

[24] 张忠志, 胡锡炎, 张磊. 线性流形上反埃尔米特广义汉密尔顿矩阵的最小二乘问题与最佳逼近问题. 数学物理学报, 2006, 26A(6): 978--986

[25] 于蕾, 张凯院, 史忠科. 线性流形上反对称正交对称矩阵反问题的最小二乘解. 数学物理学报, 2006, 26A: 1031--1038

[26] Trench W F. Hermitian R-symmetric, and Hermitian R-skew symmetric procrustes problems. Linear Algebra Appl, 2004, 387: 83--98

[27] Trench W F. Inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry. Linear Algebra Appl, 2004, 380: 199--211

[28] Trench W F. Minimization problems for (R,S)-symmetric and $(R,S)$-skew symmetric matrices. Linear
Algebra Appl, 2004, 389: 23--31

[29] 张磊,谢冬秀. 一类逆特征值问题.数学物理学报, 1993, 13A(1): 94--99

[30] Aubin J P. Applied Functional Analysis. New York: John Wiley, 1979