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Mathematics > Algebraic Geometry

arXiv:2401.00866 (math)
[Submitted on 29 Dec 2023 (v1), last revised 10 May 2024 (this version, v2)]

Title:Conditions for eigenvalue configurations of two real symmetric matrices: a signature approach

Authors:Hoon Hong, Daniel Profili, J. Rafael Sendra
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Abstract:For two real symmetric matrices, their eigenvalue configuration is the arrangement of their eigenvalues on the real line. In this paper, we provide quantifier-free necessary and sufficient conditions for two symmetric matrices to realize a given eigenvalue configuration. The basic idea is to generate a set of polynomials in the entries of the two matrices whose roots can be counted to uniquely determine the eigenvalue configuration. This result can be seen as ageneralization of Descartes' rule of signs to the case of two real univariate polynomials.
Comments: arXiv admin note: substantial text overlap with arXiv:2401.00089
Subjects: Algebraic Geometry (math.AG); Symbolic Computation (cs.SC)
Cite as: arXiv:2401.00866 [math.AG]
  (or arXiv:2401.00866v2 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.2401.00866

Submission history

From: Daniel Profili [view email]
[v1] Fri, 29 Dec 2023 22:24:29 UTC (13 KB)
[v2] Fri, 10 May 2024 14:45:59 UTC (13 KB)
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