For two real symmetric matrices, their eigenvalue configuration is therelative arrangement of their eigenvalues on the real line. We consider the following problem: given two parametric real symmetric matrices and an eigenvalue configuration, find a simple condition on the parameters such that the two matrices have the given eigenvalue configuration. In this paper, we develop theory and give an algorithm for this problem. The output of the algorithm is a condition written in terms of the signatures of certain related symmetric matrices.

翻译：对于两个实对称矩阵，其特征值配置是指它们的特征值在实轴上的相对排列。我们考虑以下问题：给定两个参数化的实对称矩阵及一个特征值配置，寻找参数上的简单条件，使得这两个矩阵具有给定的特征值配置。本文针对该问题发展理论并给出算法。算法的输出结果是以特定相关对称矩阵的符号表示的参数条件。