## 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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## 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 |
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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 |

### Available Versions of this Item

- On $p$th Roots of Stochastic Matrices (deposited 06 May 2011)
- On $p$th Roots of Stochastic Matrices (deposited 10 March 2009)
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- On $p$th Roots of Stochastic Matrices (deposited 10 March 2009)

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