矩阵方程XA-YB=D的广义中心对称解及其最佳逼近  

GENERALIZED CENTROSYMMETRIC SOLUTIONS OF THE MATRIX EQUATION XA-YB=D AND AN ASSOCIATED OPTIMAL APPROXIMATION

作　　者：袁永新[1] 张红晔 戴华[3] 

机构地区：[1]江苏科技大学数学系,镇江212003 [2]吉林铁道职业技术学院数学教研室,吉林132001 [3]南京航空航天大学数学系,南京210016

出　　处：《高等学校计算数学学报》2009年第1期1-10,共10页Numerical Mathematics A Journal of Chinese Universities

基　　金：国家自然科学基金资助项目(编号:10271055)

摘　　要：1 引言 首先引入一些记号．记C^n×m为^n×m复矩阵的集合．UC^n×n表示所有n阶酉矩阵的集合．In表示n阶单位矩阵．A^H和A^＋分别表示矩阵A的共轭转置及MoorePenrose广义逆．Let R ∈ C^n×n be a nontrivial generalized reflection matrix satisfying R = R^-1 ≠ ±In.G ∈ C^n×n is said to be generalized centrosymmetric if RGR = G.The set of all n × n generalized centrosymmetric matrices is denoted by GCSC^n×n. In this paper, a sufficient and necessary condition for the matrix equation XA - YB = D having a solution X, Y ∈ GCSC^n×n is derived, and the general solutions are given. The optimal approximation between given matrix （X, Y） and the offine subspace of solutions to the matrix equation is discussed, an explicit representation of the unique optimal approximation solution is presented, and a numerical example is provided.

关 键 词：广义逆 中心对称解 最佳逼近 矩阵方程 共轭转置 单位矩阵 矩阵A 酉矩阵 

分 类 号：O151.21[理学—数学] O174.41[理学—基础数学]