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
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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 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:  http://eprints.maths.manchester.ac.uk/id/eprint/1429 
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On $p$th Roots of Stochastic Matrices. (deposited 10 Mar 2009)
 On $p$th Roots of Stochastic Matrices. (deposited 06 May 2011) [Currently Displayed]
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