Conditions for eigenvalue configurations of two real symmetric matrices (symmetric polynomial approach)

Hoon Hong; Daniel Profili; J. Rafael Sendra

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arXiv · 2401.00089

Conditions for eigenvalue configurations of two real symmetric matrices (symmetric polynomial approach)

Hoon Hong·Daniel Profili·J. Rafael Sendra

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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. 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.

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Hoon Hong, Daniel Profili, J. Rafael Sendra. 2023-12-29. Conditions for eigenvalue configurations of two real symmetric matrices (symmetric polynomial approach). https://arxiv.org/abs/2401.00089

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