矩阵方程AX=B及其最小二乘问题的一类广义对称解  

A class of generalized symmetric solutions to the matrix equation AX=B and its least squares problem

作　　者：彭振赟[1] 尚邵阳 周昱洁 PENG Zhenyun;SHANG Shaoyang;ZHOU Yujie(School of Mathematics and Computational Sciences,Guilin University of Electronic Technology,Guilin 541004,China)

机构地区：[1]桂林电子科技大学数学与计算科学学院,广西桂林541004

出　　处：《桂林电子科技大学学报》2020年第1期70-76,共7页Journal of Guilin University of Electronic Technology

基　　金：国家自然科学基金(61627807);广西自然科学基金(2017GXNSFAA198248,2018GXNSFBA281192);桂林电子科技大学研究生教育创新计划(2019YCXS084)。

摘　　要：讨论了矩阵方程AX=B及其最小二乘问题的一类广义对称解。基于广义对称矩阵的结构特征,矩阵奇异值分解,矩阵的广义奇异值分解和矩阵分块降阶的方法,给出了矩阵方程有解的充分必要条件。得到了矩阵方程及其最小二乘问题的解的一般表达式,以及解集合中与任一给定矩阵的最佳逼近解的一般表达式。A class of generalized symmetric solutions to the matrix equation AX=B and its least squares problem are discussed. Based on the structural characteristics of generalized symmetric matrix, singular value decomposition of matrix, generalized singular value decomposition of matrix and the method of matrix block reduction, the necessary and sufficient conditions for the existence of solutions of matrix equations are given. The general expression of the solution of the matrix equation and its least squares problem and the general expression of the best approximation solution to any given matrix in the solution set are obtained.

关 键 词：广义对称矩阵 矩阵方程 MOORE-PENROSE广义逆 奇异值分解 广义奇异值分解 

分 类 号：O241.2[理学—计算数学]