X-simple image eigencones of tropical matrices

TL;DR

This paper studies the structure of eigencones in tropical algebra, characterizing matrices with unique eigenvector solutions within intervals, and provides practical criteria for identifying such matrices.

Contribution

It introduces the concept of X-simple image eigencones in tropical matrices and offers geometric and combinatorial characterizations along with efficient checking criteria.

Findings

Characterization of matrices with X-simple image eigencones

Geometric and combinatorial descriptions provided

Efficient criteria for special cases derived

Abstract

We investigate max-algebraic (tropical) one-sided systems $A\otimes x=b$ where $b$ is an eigenvector and $x$ lies in an interval $X$. A matrix $A$ is said to have $X$-simple image eigencone associated with an eigenvalue $\lambda$, if any eigenvector $x$ associated with $\lambda$ and belonging to the interval $X$ is the unique solution of the system $A\otimes y=\lambda x$ in $X$. We characterize matrices with $X$-simple image eigencone geometrically and combinatorially, and for some special cases, derive criteria that can be efficiently checked in practice.