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PDF(624 KB)
PDF(624 KB)
线性矩阵方程组AX=B,YA=D的最小二乘(R,Sσ)-交换解
The Least-square Solutions to the Linear Matrix Equations AX=B, YA=D with (R,Sσ)-commutative Matrices
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.
(R / S&sigma / )-交换矩阵 / 最小二乘解 / 最佳逼近解 {{custom_keyword}} /
(R,Sσ)-commutative matrices / least squares solution / optimal approximate {{custom_keyword}} /
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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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