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2009.21: On $p$th Roots of Stochastic Matrices

2009.21: Nicholas J. Higham and Lijing Lin (2009) On $p$th Roots of Stochastic Matrices.

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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 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2009.21
Deposited By:Nick Higham
Deposited On:10 March 2009

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