A Markov chain representation of the Perron-Frobenius eigenvector

TL;DR

This paper presents a novel Markov chain-based formula for computing the Perron-Frobenius eigenvector of a primitive matrix, generalizing the classical invariant measure formula for Markov chains.

Contribution

It introduces a new representation linking the Perron-Frobenius eigenvector to Markov chain realizations, extending classical invariant measure formulas.

Findings

Provides a formula for the eigenvector in terms of Markov chain realizations

Generalizes the classical invariant measure formula

Offers a new perspective on eigenvector computation

Abstract

We consider the problem of finding the Perron-Frobenius eigenvector of a primitive matrix. Dividing each of the rows of the matrix by the sum of the elements in the row, the resulting new matrix is stochastic. We give a formula for the Perron-Frobenius eigenvector of the original matrix, in terms of a realization of the Markov chain defined by the associated stochastic matrix. This formula is a generalization of the classical formula for the invariant probability measure of a Markov chain.