Infinitely divisible nonnegative matrices, $M$-matrices, and the embedding problem for finite state stationary Markov Chains

TL;DR

This paper explores the mathematical relationships between $M$-matrices, nonnegative roots, and the embedding problem in finite-state Markov chains, providing new characterizations and answering longstanding questions using basic matrix analysis.

Contribution

It establishes a connection between $M$-matrices and nonnegative roots, and offers a new characterization of embeddable stochastic matrices, solving a question posed over 30 years ago.

Findings

Characterization of nonnegative matrices with nonnegative roots.

Closure of matrices with matrix roots in $\\mathcal{IM}$.

New insights into the embedding problem for Markov chains.

Abstract

This paper explicitly details the relation between $M$-matrices, nonnegative roots of nonnegative matrices, and the embedding problem for finite-state stationary Markov chains. The set of nonsingular nonnegative matrices with arbitrary nonnegative roots is shown to be the closure of the set of matrices with matrix roots in $\mathcal{IM}$. The methods presented here employ nothing beyond basic matrix analysis, however it answers a question regarding $M$-matrices posed over 30 years ago and as an application, a new characterization of the set of all embeddable stochastic matrices is obtained as a corollary.