Column sums and the conditioning of the stationary distribution for a stochastic matrix
Steve Kirkland · Operators and Matrices · 2010
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 c T 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 c T } . We then characterise those vectors c T such that (T ) is bounded as T ranges over the irreducible matrices in S (c) ; for those column sum vectors c T for which is bounded, we give an upper bound on in terms of the entries in c T , and characterise the equality case.