For two real symmetric matrices, their eigenvalue configuration is the arrangement of their eigenvalues on the real line. We study the problem of determining a quantifier-free necessary and sufficient condition for two real symmetric matrices to realize a given eigenvalue configuration as a generalization of Descartes' rule of signs. We exploit the combinatorial properties of our definition for eigenvalue configuration to reduce a two-polynomial root counting problem into several single-polynomial root counting problems of symmetric polynomials. We then leverage the fundamental theorem of symmetric polynomials to derive a final quantifier-free necessary and sufficient condition for two real symmetric matrices to realize a given eigenvalue configuration.

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直线上的排列方式。作为笛卡尔符号法则的推广，我们研究如何确定两个实对称矩阵实现给定特征值配置的无量词充要条件。通过利用特征值配置定义中的组合性质，我们将双多项式根计数问题转化为多个对称多项式单多项式根计数问题。随后借助对称多项式基本定理，推导出两个实对称矩阵实现给定特征值配置的最终无量词充要条件。