Algebraic Graph Theory: Steve Kirkland

Title: Eigenvalues for stochastic matrices with a prescribed stationary distribution

Abstract: A square nonnegative matrix T is called stochastic if all of its row sums are equal to 1. Under mild conditions, it turns out that there is a positive row vector w^T (called the stationary distribution for T) whose entries sum to 1 such that the powers of T converge to the outer product of w^T with the all-ones vector. Further, the nature of that convergence is governed by the eigenvalues of T.

In this talk we explore how the stationary distribution for a stochastic matrix exerts an influence on the corresponding eigenvalues. We do so by considering the region in the complex plane comprised of all eigenvalues of all stochastic matrices with a given stationary distribution. We establish a few properties of that region, and of the variant that arises by considering the so-called reversible stochastic matrices. For the reversible version of the problem, the graphs associated with the reversible stochastic matrices are a useful tool.