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The perturbation bound for the Perron vector of a transition probability tensor
Author(s) -
Li Wen,
Cui LuBin,
Ng Michael K.
Publication year - 2013
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1886
Subject(s) - mathematics , bounded function , perturbation (astronomy) , eigenvalues and eigenvectors , uniqueness , stochastic matrix , upper and lower bounds , norm (philosophy) , tensor (intrinsic definition) , mathematical analysis , pure mathematics , combinatorics , markov chain , physics , quantum mechanics , statistics , political science , law
SUMMARY In this paper, we study the perturbation bound for the Perron vector of an m th‐order n ‐dimensional transition probability tensor P = (pi 1 , i 2 , … , i m) withpi 1 , i 2 , … , i m⩾ 0 and∑i 1 = 1 npi 1 , i 2 , … , i m= 1 . The Perron vector x associated to the largest Z ‐eigenvalue 1 of P , satisfies P x m − 1 = x where the entries x i of x are non‐negative and∑ i = 1 nx i = 1 . The main contribution of this paper is to show that when P is perturbed to an another transition probability tensorP ̃ by Δ P , the 1‐norm error between x andx ̃ is bounded by m , Δ P , and the computable quantity related to the uniqueness condition for the Perron vectorx ̃ ofP ̃ . 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. Copyright © 2013 John Wiley & Sons, Ltd.