对称正交对称矩阵逆特征值问题  

THE INVERSE EIGENVALUE PROBLEM OF SYMMETRIC ORTHO-SYMMETRIC MATRICESTHE INVERSE EIGENVALUE PROBLEM OF SYMMETRIC ORTHO-SYMMETRIC MATRICES

作　　者：胡锡炎[1] 张磊 周富照[3] 

机构地区：[1]湖南大学应用数学系,长沙410082 [2]湖南省计算中心,长沙410012 [3]长沙交通学院,长沙410076

出　　处：《计算数学》2003年第1期13-22,共10页Mathematica Numerica Sinica

基　　金：国家自然科学基金资助项目(19871024)

摘　　要：1.引言 逆特征值问题在结构动力学、分子光谱学、结构设计、参数识别和自动控制等许多领域都有重要应用,例如逆特征值方法是结构动态设计(见[10])和飞行器设计中的振动设计的有力工具,关于逆特征值问题的研究已取得了许多有意义的成果(见[1,9]).[4],[5],[3]分别就对称矩阵和双对称矩阵进行了研究,本文将就具有某种对称结构的矩阵--对称正交对称矩阵的逆特征值问题进行探讨.Let P∈ Rn×n such that PT = P, P-1 = PT.A∈Rn×n is termed symmetric orthogonal symmetric matrix ifAT = A, (PA)T = PA.We denote the set of all n × n symmetric orthogonal symmetric matrices byThis paper discuss the following two problems:Problem I. Given X ∈ Rn×m, A = diag(λ1,λ 2, ... ,λ m). Find A SRnxnP such thatAX =XAProblem II. Given A ∈ Rnδn. Find A SE such thatwhere SE is the solution set of Problem I, ||·|| is the Frobenius norm. In this paper, the sufficient and necessary conditions under which SE is nonempty are obtained. The general form of SE has been given. The expression of the solution A* of Problem II is presented. We have proved that some results of Reference [3] are the special cases of this paper.

关 键 词：对称正交对称矩阵 逆特征值 MOORE-PENROSE广义逆 FROBENIUS范数 

分 类 号：O151.21[理学—数学]