一类特殊矩阵的逆特征值问题

Inverse Eigenvalue Problem of a Special Kind of Matrices

本文主要讨论如下形式矩阵的逆特征值问题:即对给定n个实数λ_1>λ_2>…>λ_2与n-1个实数μ_1>μ_2>…>μ_(n-1),满足λ_1>μ_1>λ_2>…>λ_(n-1)>μ_(n-1)>λ_n,在α_2>α_3>…>α_(n-1)的条件下,存在唯一的一个矩阵A_n是以λ_i为其特征值;且其截边矩阵的特征值为μ_1,μ_2,…,μ_(n-1).

In this paper, we discuss the inverse eigenvalue problem of matrices as follow: Let in which α_i(i=l,2,…,n)be real, β_i(i=1,2,…,n-1)>0. Give the sequences λ:= (λ_i)_1~n and μ:=(μ_i)_1^(n-1) with λ_i>μ_i>λ_(i+1),i=1,2,…, n-1, under the condition α_2>α_3>…>α_(n-1), it is proved that there exists an unique matrix A_n which has λ_1, λ_2,…, λ_n as its eigenvalues and μ_1, μ_2, …,μ_(n-1) as the eigenvalues of A(n-1), in which and the method of how to get the solution is given here.