Spectral Properties of Positive Definite Matrices over Symmetrized Tropical Algebras and Valued Ordered fields

TL;DR

This paper explores the spectral properties of positive definite matrices within symmetrized tropical algebra and valued ordered fields, revealing how eigenvalues and eigenvectors relate across these mathematical frameworks.

Contribution

It establishes the correspondence between eigenvalues and eigenvectors in symmetrized tropical algebra and valued fields, providing new spectral insights and combinatorial formulas.

Findings

Eigenvalues of positive definite matrices match their diagonal entries in the tropical setting.

Valuations of eigenvalues over nonarchimedean fields align with tropical eigenvalues.

Under generic conditions, eigenvectors correspond via valuation and sign to tropical eigenvectors.

Abstract

We investigate the properties of positive definite and positive semi-definite symmetric matrices within the framework of symmetrized tropical algebra, an extension of tropical algebra adapted to ordered valued fields. We focus on the eigenvalues and eigenvectors of these matrices. We prove that the eigenvalues of a positive (semi)-definite matrix in the tropical symmetrized setting coincide with its diagonal entries. Then, we show that the images by the valuation of the eigenvalues of a positive definite matrix over a valued nonarchimedean ordered field coincide with the eigenvalues of an associated matrix in the symmetrized tropical algebra. Moreover, under a genericity condition, we characterize the images of the eigenvectors under the map keeping track both of the nonarchimedean valuation and sign, showing that they coincide with tropical eigenvectors in the symmetrized algebra. These results offer new insights into the spectral theory of matrices over tropical semirings, and provide combinatorial formul\ae\ for log-limits of eigenvalues and eigenvectors of parametric families of real positive definite matrices.