Characterizing matrices with $X$-simple image eigenspace in max-min semiring

TL;DR

This paper provides a combinatorial characterization of matrices with a unique eigenvector solution within a specified interval in max-min semiring, advancing understanding in tropical linear algebra and interval analysis.

Contribution

It introduces a novel combinatorial criterion for identifying matrices with $X$-simple image eigenspaces in max-min algebra, linking to stability and robustness concepts.

Findings

Characterization of matrices with $X$-simple image eigenspace

Connection to tropical linear algebra and interval analysis

Advancement in understanding of stability in max-min algebra

Abstract

A matrix $A$ is said to have $X$-simple image eigenspace if any eigenvector $x$ belonging to the interval $X=\{x\colon \underline{x}\leq x\leq\overline{x}\}$ is the unique solution of the system $A\otimes y=x$ in $X$. The main result of this paper is a combinatorial characterization of such matrices in the linear algebra over max-min (fuzzy) semiring. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability) that have been studied in max-min and max-plus algebras.