Conditions for eigenvalue configurations of two real symmetric matrices (symmetric polynomial approach)
Hoon Hong, Daniel Profili, J. Rafael Sendra
TL;DR
The paper addresses the problem of determining when the eigenvalue configuration $EC(F,G)$ of a pair of parametric real symmetric matrices $(F,G)$ matches a prescribed pattern, under the genericity assumption that $F$ and $G$ do not share eigenvalues. It develops a real-root counting framework that reduces the problem to counting positive roots of symmetric polynomials $h_r$ in the eigenvalues, then expresses these counts in terms of the coefficients of the characteristic polynomials via the Fundamental Theorem of Symmetric Polynomials and finally applies Descartes' rule of signs to obtain a quantifier-free condition. The main result states that $EC(F,G) = C_{\mathrm{sym}}\,A_{\mathrm{sym}}(F,G)$, with $C_{\mathrm{sym}} = T_m^{-1}$ depending only on the matrix size $m$ and $A_{\mathrm{sym}}(F,G) = (v(D_1),\dots,v(D_m))^{\top}$ where each $D_r$ is a polynomial in the coefficients of the characteristic polynomials of $F$ and $G$. This yields a practical, parametric criterion for eigenvalue arrangements, connecting classical root-counting methods to matrix eigenvalue configurations and enabling applications in perturbation analyses and structured updates.
Abstract
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.