Inverse M-Matrices and Ultrametric Matrices

  • Claude Dellacherie
  • Servet Martinez
  • Jaime San Martin
Part of the Lecture Notes in Mathematics book series (LNM, volume 2118)

Table of contents

  1. Front Matter
    Pages i-x
  2. Claude Dellacherie, Servet Martinez, Jaime San Martin
    Pages 1-3
  3. Claude Dellacherie, Servet Martinez, Jaime San Martin
    Pages 5-55
  4. Claude Dellacherie, Servet Martinez, Jaime San Martin
    Pages 57-84
  5. Claude Dellacherie, Servet Martinez, Jaime San Martin
    Pages 85-117
  6. Claude Dellacherie, Servet Martinez, Jaime San Martin
    Pages 119-163
  7. Claude Dellacherie, Servet Martinez, Jaime San Martin
    Pages 165-213
  8. Back Matter
    Pages 215-238

About this book

Introduction

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.

Keywords

15B48;60J45;15B51;05C50;31C20. Discrete potentials Hadamard product Inverse M-matrices Markov chains Ultrametricity and tree matrices

Authors and affiliations

  • Claude Dellacherie
    • 1
  • Servet Martinez
    • 2
  • Jaime San Martin
    • 3
  1. 1.Laboratoire Raphael Salem, UMR 6085.Universite de RouenRouenFrance
  2. 2.CMM-DIM, FCFMUniversidad de ChileSantiagoChile
  3. 3.CMM-DIM, FCFMUniversidad de ChileSantiagoChile

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-10298-6
  • Copyright Information Springer International Publishing Switzerland 2014
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-10297-9
  • Online ISBN 978-3-319-10298-6
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • About this book