Inverse M-Matrices and Ultrametric Matrices

by Claude Dellacherie, Servet Martinez, Jaime San Martin

Overview

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.

Product Details

ISBN-13:	9783319102979
Publisher:	Springer International Publishing
Publication date:	11/15/2014
Series:	Lecture Notes in Mathematics , #2118
Edition description:	2014
Pages:	236
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

Inverse M - matrices and potentials.- Ultrametric Matrices.- Graph of Ultrametric Type Matrices.- Filtered Matrices.- Hadamard Functions of Inverse M - matrices.- Notes and Comments Beyond Matrices.- Basic Matrix Block Formulae.- Symbolic Inversion of a Diagonally Dominant M - matrices.- Bibliography.- Index of Notations.- Index.