Given two real symmetric matrices, their eigenvalue configuration is the relative arrangement of their eigenvalues on the real line. In this paper, we consider the following problem: given two parametric real symmetric matrices and an eigenvalue configuration, find a simple condition on the parameters such that their eigenvalues have the given configuration. We give an algorithm which expresses the eigenvalue configuration problem as a real root counting problem of certain symmetric polynomials, whose roots can be counted using the Fundamental Theorem of Symmetric Polynomials and Descartes' rule of signs.

翻译：给定两个实对称矩阵，它们的特征值构型是指其特征值在实直线上的相对排列。本文考虑以下问题：给定两个参数化的实对称矩阵和一个特征值构型，寻找关于参数的简单条件，使得其特征值具有给定的构型。我们给出了一种算法，将特征值构型问题转化为特定对称多项式的实根计数问题，这些根可通过对称多项式基本定理和笛卡尔符号法则进行计数。