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. In this paper, we consider the problem under a mild condition that the two matrices do not share any eigenvalues. 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，其特征值构型是指它们的特征值在实轴上的相对排列。本文考虑以下问题：给定两个参数化实对称矩阵及一个特征值构型，寻求参数满足的简明条件，使得两矩阵的特征值具有给定构型。本文在"两矩阵无公共特征值"这一温和条件下研究该问题。我们提出一种算法，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，并利用对称多项式基本定理与笛卡尔符号法则实现实根计数。