Eigenvalues for stochastic matrices with a prescribed stationary distribution

Abstract

Given a vector $0<w \in \mathbb{R}^n$ whose entries sum to $1$, the region $\sigma_\mathcal{S}(w)$ in the complex plane consisting of all eigenvalues of all stochastic matrices having $w^\top$ as a left Perron vector is considered. Some general observations about this region are made, it is proven that $\bigcap_{w \in \mathbb{R}^n, w>0, w^\top \mathbf{1} =1} \sigma_\mathcal{S}(w) =[0,1],$ and a characterization is given of the vectors $w$ such that $\sigma_\mathcal{S}(w)$ contains an element $\lambda \ne 1$ with $|\lambda|=1.$ The corresponding problem for reversible stochastic matrices with given left Perron vector is also considered, as is the corresponding region $\sigma_\mathcal{R}(w),$ which is a subset of $[-1,1].$ Under a mild hypothesis on $w, $ it is proven that the smallest element of $\sigma_\mathcal{R}(w)$ corresponds to a reversible stochastic matrix whose graph is a tree with a loop at one vertex. A general lower bound on the eigenvalues of reversible stochastic matrices with given left Perron vector is also given, as is a complete description of $\sigma_\mathcal{R}(w)$ when $w$ has two or three entries.