2009.21: On $p$th Roots of Stochastic Matrices

2009.21: Nicholas J. Higham and Lijing Lin (2011) On $p$th Roots of Stochastic Matrices. Linear Algebra and its Applications, 435 (3). pp. 448-463. ISSN 1749-9097

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DOI: 10.1016/j.laa.2010.04.007

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.

Available Versions of this Item

• On $p$th Roots of Stochastic Matrices (deposited 06 May 2011) [Currently Displayed]
  • On $p$th Roots of Stochastic Matrices (deposited 10 March 2009)

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