线性矩阵方程的斜,Hermit{P,k+1},Hamilton解

doi:10.3969/j.issn.1000-5641.2018.04.004

华东师范大学学报(自然科学版) ›› 2018, Vol. 2018 ›› Issue (4): 32-46,58.doi: 10.3969/j.issn.1000-5641.2018.04.004

线性矩阵方程的斜,Hermit{P,k+1},Hamilton解

雍进军^1,2, 陈果良², 徐伟孺²

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

收稿日期:2017-07-19 出版日期:2018-07-25 发布日期:2018-07-19
通讯作者: 陈果良,男,教授,博士生导师,研究方向为数值代数.E-mail:glchen@math.ecnu.edu.cn E-mail:glchen@math.ecnu.edu.cn
作者简介:雍进军,男,副教授,研究方向为计算数学.E-mail:yongjinjun@126.com.
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^1,2, CHEN Guo-liang², XU Wei-ru²

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:2017-07-19 Online:2018-07-25 Published:2018-07-19

摘要/Abstract

摘要： 给定矩阵，P∈ C^n×n且P^*=-P=P^k+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∈ C^n×n，给出了使得Frobenius范数||Ã-ā||取得最小值的最佳逼近解ā∈ C^n×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

Abstract: Given P ∈C^n×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 Ã ∈ C^n×n, we present the optimal approximate solution ā ∈ C^n×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

中图分类号:

O241.6

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.

YONG Jin-jun, CHEN Guo-liang, XU Wei-ru. The skew-Hermitian {P,k+1} Hamiltonian solutions of a linear matrix equation[J]. Journal of East China Normal University(Natural Sc, 2018, 2018(4): 32-46,58.

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线性矩阵方程的斜,Hermit{P,k+1},Hamilton解

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

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