线性流形上对称正交对称矩阵反问题的最小二乘解  

Least-squares Solution of Inverse Problem for Orthogonal Symmetric Matrices on Linear Manifold

作　　者：臧正松[1] 

机构地区：[1]江苏科技大学数理学院,江苏镇江212005

出　　处：《江苏科技大学学报（自然科学版）》2005年第6期30-35,共6页Journal of Jiangsu University of Science and Technology:Natural Science Edition

摘　　要：研究了以下问题:问题Ⅰ:给定X,B∈Rn×m,求A∈S,使得f(A)=‖AX-B ‖=min,其中S={A∈SRn×nP| AY=C,Y,C∈Rn×m}为非空流形.问题Ⅱ:给定(A)∈Rn×n,求(A)∈SE,使得‖(A)-(A)‖=min ‖A-(A)‖,其中SE是问题Ⅰ的解集.A∈SE首先讨论了S非空的充要条件,并给出了其显式表示;其次研究了在线性流形S上反问题的最小二乘解及其最佳逼近,得到了问题Ⅰ的解和问题Ⅱ的唯一解.The following problems are considered. Problem Ⅰ： Given matrix X,B∈R^（n×m）, find A E S, such that f（A）=｜｜ AX-B｜｜=min,where non-empty set S={A∈SRp^（m×n）｜AY=C,Y,C∈R^（n×m）} is linear manifold. Problem Ⅱ ： Given matrix A^^∈R（m×n）, find A^～∈SE, such that ｜｜A^～-A^^｜｜=min A∈SE｜｜A-A^^｜｜ , where SE is the solution set of problem Ⅰ. Firstly, a necessary and sufficient condition for S to be non-empty is discussed, and the explicit representation of S is given. Secondly, the least-squares solutions of inverse problem for orthogonal symmetric matrices as well as the optimal approximation are investigated. By using the method of matrix singular value decomposition, the expression of general solution of problems Ⅰ- and the unique solution of problem Ⅱ are obtained.

关 键 词：正交对称 对称矩阵 反问题 最小二乘解 

分 类 号：O241.6[理学—计算数学]