次对称阵的逆特征值问题  

Inverse Eigenvalue Problems for Persymmetric Matrices

作　　者：王福义[1] 卢琳璋[1] 

机构地区：[1]厦门大学数学科学学院,福建厦门361005

出　　处：《厦门大学学报（自然科学版）》2004年第6期741-744,共4页Journal of Xiamen University：Natural Science

基　　金：国家自然科学基金(10271099)资助

摘　　要：讨论了3种次对称阵的逆特征值问题,其中一种是由部分特征值与部分特征向量来构造次对称阵并给出解存在的充要条件与解的表达式;另外两种是次对称阵的最佳逼近问题,分别给出其解的表达式;在每个问题证明求解过程中,本文充分利用特殊变换矩阵S,使比较复杂的次对称矩阵问题转化成熟悉对称矩阵问题来解决.A=(a_(ij))∈R^(n×n) is termed persymmetric matrix if a_(ij)=a_(n-j+1,n-i+1),i,j=1,2,…n.We denote the set of all n×n persymmetric matrices by PSR^(n×n).In this paper,We discuss the following three problems: Problems I:Given X∈R^(n×m)and Λ=diag(λ_1,λ_2,…,λ_m)∈R~^(m×m) find A∈PSR^(n×n) such that AX=XΛ.Problems II:Given X,B∈R^(n×m),find A∈PSR^(n×n)such that ‖AX-B‖=min.Problems III:Given A~*∈R^(n×n),find A∈S_E such that ‖A~*-‖=(inf)A∈S_E‖A~*-A‖.where‖.‖is Frobenius norm and S_E is the solution set of problem II.In this paper,the necessary and sufficient conditions have been given for problem I.For three problems,the explicit expression of the solutions has also been given.The uniqueness of solution,an algorithm and example are provided for the problem III.By using the matrix S,we simplify three procedures.Some examples are aslo provided.

关 键 词：对称阵 逆特征值问题 次对称矩阵 最佳逼近 表达式 充要条件 特征向量 变换矩阵 

分 类 号：O151.21[理学—数学] O241.6[理学—基础数学]