The study of M-matrices their inverses and discrete potential theory is now a well-established
part of linear algebra and the theory of Markov chains. The main focus of this monograph is the
so-called inverse M-matrix problem which asks for a characterization of nonnegative matrices
whose inverses are M-matrices. We present an answer in terms of discrete potential theory based
on the Choquet-Deny Theorem. A distinguished subclass of inverse M-matrices is ultrametric
matrices which are important in applications such as taxonomy. Ultrametricity is revealed to
be a relevant concept in linear algebra and discrete potential theory because of its relation
with trees in graph theory and mean expected value matrices in probability theory. Remarkable
properties of Hadamard functions and products for the class of inverse M-matrices are developed
and probabilistic insights are provided throughout the monograph.
