By Claude Dellacherie, Servet Martinez, Jaime San Martin
The learn of M-matrices, their inverses and discrete power conception is now a well-established a part of linear algebra and the speculation of Markov chains. the focus of this monograph is the so-called inverse M-matrix challenge, which asks for a characterization of nonnegative matrices whose inverses are M-matrices. We current a solution when it comes to discrete power idea in keeping with the Choquet-Deny Theorem. A unusual subclass of inverse M-matrices is ultrametric matrices, that are vital in purposes equivalent to taxonomy. Ultrametricity is published to be a appropriate idea in linear algebra and discrete capability conception as a result of its relation with timber in graph thought and suggest anticipated price matrices in likelihood idea. awesome homes of Hadamard capabilities and items for the category of inverse M-matrices are constructed and probabilistic insights are supplied in the course of the monograph.
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Extra resources for Inverse M-Matrices and Ultrametric Matrices
Example text
Therefore, A DA 1 ½D ½ 1 0 %½D Â ½ %0 ½ Â ½ D U J 0; showing also the desired relation between the right equilibrium potentials. 2 Inverse M -Matrices and Potentials 38 Now, we have to prove that if U is a Markov potential then A is also a Markov potential. We need to show the diagonal elements of A 1 are bounded by 1. 6) because Aii 1 D ii ii Ä 1. Â Ä ! 33 The probabilistic insight of this result is the following. Xn / killed at @. Ym / by recording only the visits to J . Ym / is also a killed Markov chain and it is not difficult to see that for i; j 2 J the mean number of visits to j starting from i is the same for both chains.
Let us start with the following result. 20 Assume P is a nonnegative matrix, whose row sums are bounded by 1. Then (i) If I P is nonsingular then P is substochastic, that is, at least one row sum is smaller than one. I P / 1 D P m m 0P . (ii) If P is irreducible and substochastic then I P is nonsingular. (iii) I P is nonsingular if and only if every i 2 I is connected to a site were P loses mass or i itself loses mass. Proof As a general remark recall that the Perron-Frobenius eigenvalue of a substochastic matrix cannot exceed one.
Proof The matrix P is constructed with the aid of the strong Markov property. T < 1; YT D j /; which is the probability that the first return to I , starting at i happens at site j . I I; P /U , from where the result follows. 3 Probabilistic Interpretation of M -Matrices and Their Inverses 27 P We remark that U D m 0 P m and therefore the killed Markov chain, associated with the substochastic matrix P , has U as its potential matrix. Thus, there is a one-to-one correspondence between potential matrices of transient Markov chains and inverses of row diagonally dominant M -matrices of the form I P where P is a substochastic matrix.