A Markov representation of Perron-Frobenius eigenvector for infinite non-negative matrix and Metzler-matrix

TL;DR

This paper introduces a method to represent the Perron-Frobenius eigenvector of infinite non-negative matrices and Metzler matrices using associated Markov chains, providing a probabilistic perspective on spectral properties.

Contribution

It offers a novel Markov chain-based representation for the Perron-Frobenius eigenvector in the context of infinite matrices, extending existing finite-dimensional results.

Findings

Provides a probabilistic representation of eigenvectors for infinite matrices.

Extends Perron-Frobenius theory to Metzler matrices using Markov chains.

Facilitates analysis of spectral properties via Markov process techniques.

Abstract

We will represent the so-called Perron-Frobenius eigenvector (if exists) for infinite non-negative matrix $A$ and Metzler matrix by using its corresponding Markov chain with probability transition function.