Tropical Analysis of the Asymptotics of the Perron-Frobenius Eigenvector

TL;DR

This paper explores how tropical analysis can be applied to understand the asymptotic behavior of eigenvectors of a broader class of matrices, extending known results and providing new characterizations.

Contribution

It generalizes the connection between asymptotic eigenvector behavior and tropical matrices beyond classical cases, offering new insights and characterizations.

Findings

Extended tropical analysis to a larger class of matrices

Provided a complete characterization for certain matrices

Identified limitations in generalizing classical results

Abstract

Asymptotic properties of matrices are, in general, difficult to analyze with classical mathematical techniques. In very specific cases, there is a well-known connection between the asymptotic behavior of a matrix's leading eigenvector and the corresponding "tropical" matrix, arising out of the $max$ and $min$ operations innate in tropical analysis. In this paper we examine a more general class of matrices, and explore the extent to which we can generalize the results using tropical techniques. We find that while the original results do not easily generalize, we can still make some useful statements about the asymptotic behavior in the general case, and can give a complete characterization for a larger class of matrices than previously examined.