Tropical totally positive matrices

TL;DR

This paper explores the tropical analogues of totally positive matrices, revealing their structure, factorization, and eigenvalues through nonarchimedean valuation and polyhedral representations.

Contribution

It establishes the correspondence between tropical totally positive matrices and Monge matrices, providing explicit polyhedral descriptions and eigenvalue relations.

Findings

Nonarchimedean valuation maps totally positive matrices to Monge matrices.

Tropical totally nonnegative matrices with finite permanent can be factorized into elementary matrices.

Eigenvalues of tropical totally nonnegative matrices relate to those over nonarchimedean fields.

Abstract

We investigate the tropical analogues of totally positive and totally nonnegative matrices. These arise when considering the images by the nonarchimedean valuation of the corresponding classes of matrices over a real nonarchimedean valued field, like the field of real Puiseux series. We show that the nonarchimedean valuation sends the totally positive matrices precisely to the Monge matrices. This leads to explicit polyhedral representations of the tropical analogues of totally positive and totally nonnegative matrices. We also show that tropical totally nonnegative matrices with a finite permanent can be factorized in terms of elementary matrices. We finally determine the eigenvalues of tropical totally nonnegative matrices, and relate them with the eigenvalues of totally nonnegative matrices over nonarchimedean fields.