Combinatorial Types of Tropical Eigenvectors

TL;DR

This paper investigates the piecewise linear structure of the tropical eigenvector map, identifying its cones of linearity and exploring their properties, with applications to statistical ranking.

Contribution

It characterizes the cones of linearity of the tropical eigenvector map and analyzes their structure, especially for skew-symmetric matrices, advancing understanding of tropical eigenvector geometry.

Findings

Cones of linearity are simplicial but do not form a fan.

The structure of these cones is explicitly determined.

Application to statistical ranking is explored.

Abstract

The map which takes a square matrix to its tropical eigenvalue-eigenvector pair is piecewise linear. We determine the cones of linearity of this map. They are simplicial but they do not form a fan. Motivated by statistical ranking, we also study the restriction of that cone decomposition to the subspace of skew-symmetric matrices.