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

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 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.