Conditions for eigenvalue configurations of two real symmetric matrices (signature approach)
Hoon Hong, Daniel Profili, J. Rafael Sendra
TL;DR
This work addresses the problem of characterizing when two parametric real symmetric matrices $F$ and $G$ exhibit a prescribed eigenvalue configuration. It develops a structured, quantifier-free criterion by proving a main theorem: $EC(F,G) = C_{\text{sig}}\,A_{\text{sig}}(F,G)$, where $C_{\text{sig}}$ is a fixed combinatorial matrix and $A_{\text{sig}}(F,G)$ collects signatures of matrices $f_e(G)$ built from the derivatives of $f(x) = \det(xI_m - F)$; these signatures are polynomial in the parameters, yielding a quantifier-free condition. The proof proceeds through strongly generic, generic, and arbitrary cases, using Hadamard-based transforms and Descartes' rule of signs to convert eigenvalue counts into signature-based constraints. The result provides a practical, modular framework for determining eigenvalue configurations and opens avenues for optimization and simplification of the output in applications. The approach generalizes Descartes' rule of signs to multiple univariate polynomials via a combinatorial-algebraic construction grounded in matrix signatures, with potential impact in rank updates, constrained optimization, and stability analysis.
Abstract
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.