Inverse M-Matrices and Ultrametric Matrices

The book "Inverse M-Matrices and Ultrametric Matrices" offers a comprehensive examination of M-matrices, their inverses, and discrete potential theory, which is now considered an integral part of linear algebra and Markov chain theory. The focus of this monograph is on the so-called Inverse M-Matrix Problem, which aims to characterize non-negative matrices whose inverses are M-matrices. The authors present a solution based on discrete potential theory, relying on the Choquet-Deny theorem. A prominent subfield of inverse M-matrices is ultrametric matrices, which are significant in applications such as taxonomy. The relevance of ultrametricity is highlighted in linear algebra and discrete potential theory through its connection to trees in graph theory and to mean expectation matrices in probability theory. Additionally, notable properties of Hadamard functions and products for the class of inverse M-matrices are developed, while the monograph is enriched with probabilistic insights.