Higham, Nicholas J. and Lin, Lijing (2011) On $p$th Roots of Stochastic Matrices. Linear Algebra and its Applications, 435 (3). pp. 448463. ISSN 17499097
This is the latest version of this item.
PDF
paper11.pdf Download (427kB) 
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 two 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:  Article 

Uncontrolled Keywords:  Stochastic matrix, nonnegative matrix, matrix $p$th root, primary matrix function, nonprimary matrix function, PerronFrobenius 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:  06 May 2011 
Last Modified:  20 Oct 2017 14:12 
URI:  https://eprints.maths.manchester.ac.uk/id/eprint/1429 
Available Versions of this Item

On $p$th Roots of Stochastic Matrices. (deposited 10 Mar 2009)
 On $p$th Roots of Stochastic Matrices. (deposited 06 May 2011) [Currently Displayed]
Actions (login required)
View Item 