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This paper explicitly details the relation between $M$-matrices, nonnegativeroots of nonnegative matrices, and the embedding problem for finite-statestationary Markov chains. The set of nonsingular nonnegative matrices witharbitrary nonnegative roots is shown to be the closure of the set of matriceswith matrix roots in $\mathcal{IM}$. The methods presented here employ nothingbeyond basic matrix analysis, however it answers a question regarding$M$-matrices posed over 30 years ago and as an application, a newcharacterization of the set of all embeddable stochastic matrices is obtainedas a corollary.

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