| 线性矩阵方程组AX=B,YA=D的最小二乘(R,S_σ)-交换解 |
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| 引用本文： | 文娅琼,李姣芬,黎稳.线性矩阵方程组AX=B,YA=D的最小二乘(R,S_σ)-交换解[J].数学学报,2019,62(6):833-852. |
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| 作者姓名： | 文娅琼 李姣芬 黎稳 |
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| 作者单位： | 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 |
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| 基金项目： | 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） |
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| 摘 要： | Trench在Characterization and properties of (R,S_σ)-commutative matrices,Linear Algebra Appl.,2012,436:4261-4278]中给出了(R,S_σ)-交换矩阵的定义.本文在此基础上讨论(R,S_σ)-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,S_σ)-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,S_σ)-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,S_σ)-交换解存在的充要条件及其具体解析表达式.
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| 关 键 词： | (R S_σ)-交换矩阵 最小二乘解 最佳逼近解 |
The Least-square Solutions to the Linear Matrix Equations AX=B,YA=D with (R,Sσ)-commutative Matrices |
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| Ya Qiong WEN,Jiao Fen LI,Wen LI.The Least-square Solutions to the Linear Matrix Equations AX=B,YA=D with (R,Sσ)-commutative Matrices[J].Acta Mathematica Sinica,2019,62(6):833-852. |
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| Authors: | Ya Qiong WEN Jiao Fen LI Wen LI |
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| Institution: | 1. School of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin 541004, P. R. China;
2. 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;
3. School of Mathematical Sciences, South China Normal University, China 510631, P. R. China |
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| Abstract: | Trench gave the definition of the (R,Sσ)-commutative matrix inCharacterization 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. |
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| Keywords: | (R Sσ)-commutative matrices least squares solution optimal approximate |
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