华东师范大学学报（自然科学版） >

2018 , Vol. 2018 >Issue 4: 32 - 46,58

DOI: https://doi.org/10.3969/j.issn.1000-5641.2018.04.004

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
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  • 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）

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

  • YONG Jin-jun ,
  • CHEN Guo-liang ,
  • XU Wei-ru
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  • 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

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摘要

给定矩阵，P∈ Cn×n且P*=-P=Pk+1.考虑了矩阵方程，AX=B存在斜Hermite{P，k+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Ã∈ Cn×n，给出了使得Frobenius范数||Ã-ā||取得最小值的最佳逼近解ā∈ Cn×n.当矩阵方程，AX=B，不相容时，给出了斜，Hermite{P，k+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.

关键词： 斜Hermite矩阵; Hamilton矩阵; 最小二乘解; 斜Hermite{P,k+1}Hamilton矩阵

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 P ∈Cn×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.

Key words： skew-Hermitian matrix; Hamiltonian matrix; least-squares solution; skew-Hermitian{P,k+1}Hamiltonian matrix

参考文献

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