Perturbations of non-diagonalizable stochastic matrices with preservation of spectral properties
Pauwelyn,Pieter-Jan Guerry,M. A.
Pauwelyn,Pieter-Jan
Guerry,M. A.
Abstract
In Markov chain theory, stochastic matrices are used to describe inter-state transitions. Powers of such transition matrices are computed to determine the behaviour within a Markov system. For this, diagonalizable matrices are preferred because of their useful properties. The non-diagonalizable matrices are therefore undesirable. The aim is to determine a nearby diagonalizable matrix A, starting from a non-diagonalizable matrix (A) over tilde. Previous studies tackled this problem, limited to 3 x 3 stochastic matrices. In this paper, these results are generalized for n x n stochastic matrices. Spectral properties of A are preserved in this process, such that A and (A) over tilde have coinciding semisimple eigenvalues and coinciding corresponding eigenvectors. This problem is examined and solved in this study and an algorithm is presented to find such a diagonalizable matrix (A) over tilde.
Description
Publisher Copyright: © 2021 Informa UK Limited, trading as Taylor & Francis Group.
Date
2022-10
Journal Title
Journal ISSN
Volume Title
Publisher
Research Projects
Organizational Units
Journal Issue
Keywords
Stochastic matrices, non-diagonalizable matrices, perturbation theory, Markov chains, EIGENVALUES
Citation
Pauwelyn, P-J & Guerry, M A 2022, 'Perturbations of non-diagonalizable stochastic matrices with preservation of spectral properties', Linear & Multilinear Algebra, vol. 70, no. 20, pp. 5115-5145. https://doi.org/10.1080/03081087.2021.1904813