数学

线性矩阵方程的斜,Hermit{P,k+1},Hamilton解

  • 雍进军 ,
  • 陈果良 ,
  • 徐伟孺
展开
  • 1. 贵州师范学院 数学与计算机科学学院, 贵阳 550018;
    2. 华东师范大学 数学科学学院, 上海 200241
雍进军,男,副教授,研究方向为计算数学.E-mail:yongjinjun@126.com.

收稿日期: 2017-07-19

  网络出版日期: 2018-07-19

基金资助

国家自然科学基金,(11471122);2016,年度贵州省科技平台及人才团队专项基金项目,(黔科合平台人才【2016】5609);贵州师范学院校级课题,(2015BS009)

The skew-Hermitian {P,k+1} Hamiltonian solutions of a linear matrix equation

  • YONG Jin-jun ,
  • CHEN Guo-liang ,
  • XU Wei-ru
Expand
  • 1. Department of Mathematics and Computer Science, Guizhou Education University, Guiyang 550018, China;
    2. School of Mathematical Sciemces, East China Normal University, Shanghai 200241, China

Received date: 2017-07-19

  Online published: 2018-07-19

摘要

给定矩阵,PCn×nP*=-P=Pk+1.考虑了矩阵方程,AX=B存在斜Hermite{Pk+1}(斜)Hamilton解的充要条件,并给出了解的表达式.进一步,对于任意给定的矩阵ÃCn×n,给出了使得Frobenius范数||Ã-ā||取得最小值的最佳逼近解āCn×n.当矩阵方程,AX=B,不相容时,给出了斜,Hermite{Pk+1}(斜)Hamilton最小二乘解,在此条件下,给出了对于任意给定矩阵的最佳逼近解.最后给出一些数值实例.

本文引用格式

雍进军 , 陈果良 , 徐伟孺 . 线性矩阵方程的斜,Hermit{P,k+1},Hamilton解[J]. 华东师范大学学报(自然科学版), 2018 , 2018(4) : 32 -46,58 . DOI: 10.3969/j.issn.1000-5641.2018.04.004

Abstract

Given PCn×n and P*=-P=P k+1, we consider the necessary and sufficient conditions such that the matrix equation AX=B is consistent with the skew-Hermitian {P, k + 1} (skew-) Hamiltonian structural constraint. Then, the corresponding expressions of the constraint solutions are also obtained. For any given matrix à ∈ Cn×n, we present the optimal approximate solution ā ∈ Cn×n such that ||Ã-ā|| is minimized in the Frobenius norm sense. If the matrix equation AX=B is not consistent, its least-squares skew-Hermitian {P, k + 1} (skew-) Hamiltonian solutions are given. Under the least-square sense, we consider the best approximate solutions to any given matrix. Finally, some illustrative experiments are also presented.

参考文献

[1] BAI Z J. The solvability conditions for inverse eigenvalue problem of Hermitian and generalized skew-Hamiltonian matrices and its approximation[J]. Inverse Problems, 2003, 19(5):1185-1194.
[2] CHEN H C. Generalized reflexive matrices:Special properties and applications[J]. SIAM Journal on Matrix Analysis and Applications, 1998, 19(1):140-153.
[3] 陈景良,陈向晖. 特殊矩阵[M]. 北京:清华大学出版社, 2001.
[4] RESPONDEK J. Controllability of dynamical systems with constraints[J]. Systems & Control Letters, 2005, 54(4):293-314.
[5] LIANG M, DAI L. The left and right inverse eigenvalue problems of generalized reflexive and anti-reflexive matrices[J]. Journal of Computational and Applied Mathematics, 2010, 234(3):743-749.
[6] PENG Z Y, DENG Y B, LIU J W. Least-squares solution of inverse problem for Hermitian anti-reflexive matrices and its appoximation[J]. Acta Mathematica Sinica, 2006, 22(2):477-484.
[7] CVETKOVIC-ILIiÍC D S. The reflexive solutions of the matrix equation AXB=C[J]. Computers & Mathematics with Applications, 2006, 51(6/7):897-902.
[8] PEND Z, HU X. The reflexive and anti-reflexive solutions of the matrix equation AX=B[J]. Linear Algebra and its Applications, 2003, 375:147-155.
[9] HERRERO A, THOME N. Using the GSVD and the lifting technique to find {P, k+1} reflexive and anti-reflexive solutions of AXB=C[J]. Applied Mathematics Letters, 2011, 24(7):1130-1141.
[10] ZHANG J C, ZHOU S Z, HU X Y. The (P, Q) generalized reflexive and anti-reflexive solutions of the matrix equation AX=B[J]. Applied Mathematics and Computation, 2009, 209(2):254-258.
[11] ZHOU K, DOYLE J C, GLOVER K. Robust and Optimal Control[M]. New Jersey:Prentice hall, 1996.
[12] SIMA V. Algorithms for Linear-Quadratic Optimization[M]. New York:Marcel Dekker, 1996.
[13] XIE D, HU X, SHENG Y. The solvability conditions for the inverse eigenproblems of symmetric and generalized centro-symmetric matrices and their approximations[J]. Linear Algebra and Its Applications, 2006, 418(1):142-152.
[14] CAI J, CHEN J L. Least-squares solutions of generalized inverse eigenvalue problem over Hermitian-Hamiltonian matrices with a submatrix constraint[J]. Computational and Applied Mathematics, 2018, 37:593-603.
[15] 谢冬秀,张磊. 一类反对称问题的最小二乘解[J]. 工程数学学报. 1993, 10(4):25-34.
[16] PENG X, HU X, ZHANG L. The reflexive and anti-reflexive solutions of the matrix equation AHXB=C[J]. Journal of Computational and Applied Mathematics, 2007, 200(2):749-760.
[17] XIE D, HU X, ZHANG L. The solvability conditions for inverse eigenproblem of symmetric and anti-persymmetric matrices and its approximation[J]. Numerical Linear Algebra with Applications, 2003, 10(3):223-234.
[18] XU W R, CHEN G L, GONG Y. Procrustes problems and inverse eigenproblems for multilevel block α-circulants[J]. Numerical Linear Algebra with Applications, 2016, 23(5):906-930.
文章导航

/