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  • Book
  • © 2014

Inverse M-Matrices and Ultrametric Matrices

  • Provides a unified algebraic and probabilistic approach

  • Describes graphs and algorithms for inverse M-matrices

  • Gives examples and fields of applications of M-matrices

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2118)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-x
  2. Introduction

    • Claude Dellacherie, Servet Martinez, Jaime San Martin
    Pages 1-3
  3. Inverse M-Matrices and Potentials

    • Claude Dellacherie, Servet Martinez, Jaime San Martin
    Pages 5-55
  4. Ultrametric Matrices

    • Claude Dellacherie, Servet Martinez, Jaime San Martin
    Pages 57-84
  5. Graph of Ultrametric Type Matrices

    • Claude Dellacherie, Servet Martinez, Jaime San Martin
    Pages 85-117
  6. Filtered Matrices

    • Claude Dellacherie, Servet Martinez, Jaime San Martin
    Pages 119-163
  7. Hadamard Functions of Inverse M-Matrices

    • Claude Dellacherie, Servet Martinez, Jaime San Martin
    Pages 165-213
  8. Back Matter

    Pages 215-238

About this book

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

  • Laboratoire Raphael Salem, UMR 6085., Universite de Rouen, Rouen, France

    Claude Dellacherie

  • CMM-DIM, FCFM, Universidad de Chile, Santiago, Chile

    Servet Martinez, Jaime San Martin

Bibliographic Information

  • Book Title: Inverse M-Matrices and Ultrametric Matrices

  • Authors: Claude Dellacherie, Servet Martinez, Jaime San Martin

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-319-10298-6

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing Switzerland 2014

  • Softcover ISBN: 978-3-319-10297-9Published: 04 December 2014

  • eBook ISBN: 978-3-319-10298-6Published: 14 November 2014

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: X, 236

  • Number of Illustrations: 14 b/w illustrations

  • Topics: Potential Theory, Probability Theory, Game Theory

Buying options

eBook USD 39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 49.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions