## Abstract

In this paper, we study the perturbation bound for the Perron vector of an mth-order n-dimensional transition probability tensor P=(pi1,i2,...,im) with pi1,i2,...,im≥0 and ∑i1=1npi1,i2,...,im=1. The Perron vector x associated to the largest Z-eigenvalue 1 of P, satisfies Pxm-1=x where the entries x_{i} of x are non-negative and ∑i=1nxi=1. The main contribution of this paper is to show that when P is perturbed to an another transition probability tensor P̃ by ΔP, the 1-norm error between x and x̃ is bounded by m, ΔP, and the computable quantity related to the uniqueness condition for the Perron vector x̃ of P̃. Based on our analysis, we can derive a new perturbation bound for the Perron vector of a transition probability matrix which refers to the case of m=2. Numerical examples are presented to illustrate the theoretical results of our perturbation analysis.

Original language | English |
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Pages (from-to) | 985-1000 |

Number of pages | 16 |

Journal | Numerical Linear Algebra with Applications |

Volume | 20 |

Issue number | 6 |

DOIs | |

Publication status | Published - Dec 2013 |

## Scopus Subject Areas

- Algebra and Number Theory
- Applied Mathematics

## User-Defined Keywords

- Perron vector
- Peturbation bound
- Transition probability tensor