On $p$th Roots of Stochastic Matrices

Higham, Nicholas J. and Lin, Lijing (2009) On $p$th Roots of Stochastic Matrices. [MIMS Preprint]

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Abstract

In Markov chain models in finance and healthcare a transition matrix over a certain time interval is needed but only a transition matrix over a longer time interval may be available. The problem arises of determining a stochastic $p$th root of a stochastic matrix (the given transition matrix). By exploiting the theory of functions of matrices, we develop results on the existence and characterization of matrix $p$th roots, and in particular on the existence of stochastic $p$th roots of stochastic matrices. Our contributions include characterization of when a real matrix has a real $p$th root, a classification of $p$th roots of a possibly singular matrix, a sufficient condition for a $p$th root of a stochastic matrix to have unit row sums, and the identification of classes of stochastic matrices that have stochastic $p$th roots for all $p$. We also delineate a wide variety of possible configurations as regards existence, nature (primary or nonprimary), and number of stochastic roots, and develop a necessary condition for existence of a stochastic root in terms of the spectrum of the given matrix.

Item Type: MIMS Preprint
Uncontrolled Keywords: Stochastic matrix, nonnegative matrix, matrix $p$th root, primary matrix function, nonprimary matrix function, Perron--Frobenius theorem, Markov chain, transition matrix, embeddability problem, $M$-matrix, inverse eigenvalue problem
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Nick Higham
Date Deposited: 10 Mar 2009
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/1241

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