对称正交对称矩阵反问题的最小二乘解  

LEAST-SQUARES SOLUTIONS OF INVERSE PROBLEMS FOR SYMMETRIC ORTHOGONAL SYMMETRIC MATRICES

作　　者：戴华[1] 

机构地区：[1]南京航空航天大学数学系,南京210016

出　　处：《计算数学》2003年第1期59-66,共8页Mathematica Numerica Sinica

基　　金：国家自然科学基金(10271055);江苏省"青蓝工程"基金资助项目

摘　　要：1.引言 近几年来,对矩阵反问题AX=B的研究已取得了许多进展[1,2].最近,胡锡炎等[3]研究了对称正交对称矩阵反问题,给出了问题有解的条件.但在结构模型修正等实际问题中,矩阵X和B通常由试验给出,一般难以保证问题的解存在.因此本文讨论对称正交对称矩阵反问题的最小二乘解.Let P ∈ Rn×n be a symmetric orthogonal matrix. A∈Rn×n is called a symmetric orthogonal symmetric matrix if AT = A and (PA) T = PA. The set of all n × n symmetric orthogonal symmetric matrices is denoted by SRnxnp. This paper discusses the following problems: Problem I. Given X,B∈ Rn×m, find A ∈SRn×np such that||AX - B|| = min Problem II. Given A∈ Rn×n, find A∈SL such thatwhere ||·|| is the Frobenius norm, and SL is the solution set of Problem I.The general form of SL is given. The solvability conditions for the inverseproblem AX = B in SRn×nP are obtained. The expression of the solution toProblem II is presented.

关 键 词：对称正交对称矩阵 反问题 最小二乘解 MOORE-PENROSE广义逆 Hadamard积 FROBENIUS范数 内积空间 

分 类 号：O151.21[理学—数学]