经典代数特征值反问题的一般提法

A General Formulation for the Algebraic Inverse Eigenvalue Problem

§1.引言 [3]曾提出两类Hermiie阵的代数特征值反问题,后来被人们称之为加法问题和乘法问题并推广到更一般的情形.到目前止,经典代数特征值反问题在数学上的最一般提法如下: 问题G.给定n+1个n阶实对称矩阵A,A_1,…,A_n和n个实数λ_1,…,λ_n,求n个实数x_1,…,x_n。

In this paper the inverse eigenvalue problem for real symmetric matrices isreformulated as follows: Let A(x) be a given real n×n symmetric matrix-valued analytic functiondefined on R^m, and let λ_1~*,…,λ_p~* be p given real numbers (p≤n). Find an x~*∈R^m such that f(x~*)=min_xf(x), where f(x) = min_(π∈g_n)∑from i=1 to p(λ_(π(i))(x)-λ_i~*)~2, p_n is the set of all permutations of {1,2,…,n} and λ_1,(x),…,λ_n(x) are the eigenvalues of A(x). Some properties of f(x) are studied and a necessary condition for x~* to be alocal minimum point of f(x) is given.