Inverse M-Matrices and Ultrametric Matrices

Nonfiction, Science & Nature, Mathematics, Vector Analysis, Statistics
Cover of the book Inverse M-Matrices and Ultrametric Matrices by Claude Dellacherie, Servet Martinez, Jaime San Martin, Springer International Publishing
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Author: Claude Dellacherie, Servet Martinez, Jaime San Martin ISBN: 9783319102986
Publisher: Springer International Publishing Publication: November 14, 2014
Imprint: Springer Language: English
Author: Claude Dellacherie, Servet Martinez, Jaime San Martin
ISBN: 9783319102986
Publisher: Springer International Publishing
Publication: November 14, 2014
Imprint: Springer
Language: English

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.

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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.

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