矩阵方程AXB+CYD=E的对称极小范数最小二乘解  

LEAST SQUARES SYMMETRIC SOLUTION OF THE MATRIX EQUATION AXB+CYD=E WITH THE LEAST NORM

作　　者：袁仕芳[1] 廖安平[1] 雷渊[1] 

机构地区：[1]湖南大学数学与计量经济学院,长沙410082

出　　处：《计算数学》2007年第2期203-216,共14页Mathematica Numerica Sinica

基　　金：湖南省自然科学基金(03JJY6028)资助项目.

摘　　要：对于任意给定的矩阵A∈Rm×n,B∈Rn×s,C∈Rm×k,D∈Rk×s,E∈Rm×s,本文利用矩阵的Kmnecker积和Moore-Penrose广义逆,研究矩阵方程AXB+CYD=E的对称极小范数最小二乘解,得到了解的表达式．并由此给出了矩阵方程AXB=C的双对称极小范数最小二乘解的表达式．此外,我们还给出了求矩阵方程AXB=C的双对称极小范数最小二乘解的数值算法和数值例子．A ∈ R^n×n is called a bisymmetric matrix if A=（aij）n×n,aij = aji,aij = an＋1-i,n＋1-j （i, j = 1, 2,... , n）. The set of all n×n bisymmetric matrices is denoted by BSR^n×n. By using Moore-Penrose generalized inverse and the Kronecker product of matrices, we derive the expression of the least squares symmetric solution of the matrix equation AXB ＋CYD = E with the least norm. Based on this, we present the expression of the least squares bisymmetric solution of the matrix equation AXB = C with the least norm, and provide a numerical algorithm and numerical experiments for finding the least squares bisymmetric solution with the least norm.

关 键 词：对称矩阵 双对称矩阵 极小范数解 最小二乘解 Moore-Penrose 广义逆 KRONECKER积 

分 类 号：O241.6[理学—计算数学]