矩阵方程A^TXB+B^TX^TA=D的极小范数最小二乘解的迭代解法  

AN ITERATIVE METHOD FOR THE MINIMUM NORM-LEAST SQUARE SOLUTION OF THE MATRIX EQUATION A^TXB+B^TX^TA=D

作　　者：盛兴平[1,2] 苏友峰[2] 陈果良[2] 

机构地区：[1]阜阳师范学院数学与计算科学学院,阜阳236032 [2]华东师范大学数学系,上海200062

出　　处：《高等学校计算数学学报》2008年第4期352-362,共11页Numerical Mathematics A Journal of Chinese Universities

基　　金：上海市科技攻关项目(062112065);中国博士后科学基金项目(20060400634);安徽省高校青年教师重点科研资助项目(2006jql220)

摘　　要：1 引言 设R^m×n表示m×n实矩阵的全体，A^T表示矩阵A的转置，R（A）和N（A）分别表示矩阵A的值域和零空间，A^＋表示矩阵A的Moore—Penrose广义逆，A×B表示矩阵A与B的Kronecker乘积，In this paper, an iterative method is presented to solve the following two problems： （1） Given A ∈ R^n×m, B ∈ R^p×m, D ∈ R^m×m, find X ∈ R^n×p, such that ｜｜A^TXB ＋ B^TXTA - D ｜｜=min. （2） Given X ∈ R^n×p, find X ∈ SE, such that ｜｜ X -X ｜｜=min, where SE is the solution set of problem （1）. Using the iterative method, for any initial matrix X0, a solution X＊ of problem 1 can be obtained within finite iteration steps in absence of roundoff errors, and the minimum norm of problem 1 can also be obtained by choosing a special initial matrix. In addition, by finding the minimum norm solution X＊ of the new residual problem ｜｜ A^TXB ＋ B^TX^TA - D ｜｜=min, the solution of problem （2） can be obtained, i.e., X = X＊＋ X, where D = D- 2AXB. Finally numerical examples are shown that the iterative method is efficient.

关 键 词：极小范数最小二乘解 迭代解法 矩阵方程 KRONECKER乘积 矩阵A 实矩阵 零空间 广义逆 

分 类 号：O151.21[理学—数学] O241.6[理学—基础数学]