For two real symmetric matrices, their eigenvalue configuration is the arrangement of their eigenvalues on the real line. In this paper, we provide quantifier-free necessary and sufficient conditions for two symmetric matrices to realize a given eigenvalue configuration. The basic idea is to generate a set of polynomials in the entries of the two matrices whose roots can be counted to uniquely determine the eigenvalue configuration. This result can be seen as ageneralization of Descartes' rule of signs to the case of two real univariate polynomials.

翻译：对于两个实对称矩阵，其特征值配置是指它们特征值在实轴上的排列顺序。本文给出了两个对称矩阵实现给定特征值配置的无量词充分必要条件。基本思想是构造一组关于两个矩阵元素的多元多项式，通过计算这些多项式的根的数量可唯一确定特征值配置。该结果可视为笛卡尔符号法则在双实系数单变量多项式情形下的推广。