线性矩阵方程组AX=B,YA=D的最小二乘(R,S_σ)-交换解  

The Least-square Solutions to the Linear Matrix Equations AX=B,YA=D with(R,S_σ)-commutative Matrices

作　　者：文娅琼 李姣芬 黎稳[3] Ya Qiong WEN;Jiao Fen LI;Wen LI(School of Mathematics and Computational Science,Guilin University of Electronic Technology,Guilin 541004,P.R.China;School of Mathematics and Computational Science,Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation,Guilin University of Electronic Technology,Guilin 541004,P.R.China;School of Mathematical Sciences,South China Normal University,China 510631,P.R.China)

机构地区：[1]桂林电子科技大学数学与计算科学学院,桂林541004 [2]桂林电子科技大学数学与计算科学学院广西高校数据分析与计算重点实验室,桂林541004 [3]华南师范大学数学科学学院,广州510631

出　　处：《数学学报（中文版）》2019年第6期833-852,共20页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(11761024,11561015,11671158,U1811464);广西自然科学基金资助项目(2016GXNSFAA380074,2016GXNSFFA380009,2017GXNSFBA198082);桂林电子科技大学研究生优秀学位论文培育项目(17YJPYSS24)

摘　　要：Trench在[Characterization and properties of(R,Sσ)-commutative matrices,Linear Algebra Appl.,2012,436:4261-4278]中给出了(R,Sσ)-交换矩阵的定义.本文在此基础上讨论(R,Sσ)-交换矩阵的一般性结构,对给定的矩阵X,Y,B,D,以及线性方程组AX=B,YA=D在(R,Sσ)-交换矩阵集合中的最小二乘问题及最佳逼近问题.细致分析最小二乘(R,Sσ)-交换解和最佳逼近解的具体解析表达式.同时在方程组相容情况下分析(R,Sσ)-交换解存在的充要条件及其具体解析表达式.Trench gave the definition of the(R,Sσ)-commutative matrix in[Characterization and properties of(R,Sσ)-commutative matrices,Linear Algebra Appl.,2012,436:4261-4278].This paper discusses the general structure of(R,Sσ)-commutative matrix,and the least squares problem and the best approximation problem of linear equations AX=B,YA=D in the set of(R,Sσ)-commutative matrices for given matrix X,Y,B,D.Then we analysis the expressions of the least-square commutative solution and the best approximate solution of(R,Sσ)-commutative matrix in detail.At the same time,the necessary and sufficient conditions for the existence of the(R,Sσ)-commutative solution are analyzed when the linear matrix equations is consistent.

关 键 词：(R Sσ)-交换矩阵 最小二乘解 最佳逼近解 

分 类 号：O177.2[理学—数学]