Perturbations of non-diagonalizable stochastic matrices with preservation of spectral properties

Pauwelyn, Pieter-Jan; Guerry, Marie-Anne

doi:10.1080/03081087.2021.1904813

Cited by 1 publication

(1 citation statement)

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“…This is based on the notion that diagonalizable matrices densely fill the set of all matrices [67], meaning that it is always possible to find some nearby matrix X that is diagonalizable. In one study, a method along these lines was developed for dealing with non-diagonalizable transition matrices [68]. In particular, for a starting transition matrix P , a perturbation matrix E is found such that P = P + E preserves a number of the spectral properties of P and is diagonalizable.…”

Section: Alternative Methodsmentioning

confidence: 99%

A Tutorial on the Spectral Theory of Markov Chains

Seabrook¹,

Wiskott²

2022

Preprint

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Markov chains are a class of probabilistic models that have achieved widespread application in the quantitative sciences. This is in part due to their versatility, but is compounded by the ease with which they can be probed analytically. This tutorial provides an in-depth introduction to Markov chains, and explores their connection to graphs and random walks. We utilize tools from linear algebra and graph theory to describe the transition matrices of different types of Markov chains, with a particular focus on exploring properties of the eigenvalues and eigenvectors corresponding to these matrices. The results presented are relevant to a number of methods in machine learning and data mining, which we describe at various stages. Rather than being a novel academic study in its own right, this text presents a collection of known results, together with some new concepts. Moreover, the tutorial focuses on offering intuition to readers rather than formal understanding, and only assumes basic exposure to concepts from linear algebra and probability theory. It is therefore accessible to students and researchers from a wide variety of disciplines.

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Section: Alternative Methodsmentioning

confidence: 99%