  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.
