Abstract

For an irreducible stochastic matrix T, we consider a certain condition number (T), which measures the sensitivity of the stationary distribution vector to perturbations in T, and study the extent to which the column sum vector for T provides information on (T). Specifically, if cT is the column sum vector for some stochastic matrix of order n, we define the set S(c) = {A|A is an n × n stochastic matrix with column sum vector cT }. We then characterise those vectors cT such that (T) is bounded as T ranges over the irreducible matrices in S(c); for those column sum vectors cT for which is bounded, we give an upper bound on in terms of the entries in cT , and characterise the equality case.