Interlacing Eigenvalues in Time Reversible Markov Chains

For irreducible, reversible, finite state chains we observe that two sets of eigenvalues related to the transition rate matrix Q, (λ₀, …, λ_m) and (γ₁, …, γ_m), are interlaced so that λ₀ < γ₁ < λ₁ < ⋯ < γ_m < λ_m. Many quantities associated with ℒ_πT_A, the distribution of the first passage time to A starting in steady state, can be simply expressed in terms of these eigenvalues, and the interlacing property can be exploited to obtain approximations. There is a close connection between interlacing eigenvalues and completely monotone distributions, as well as a representation for finite mixtures of exponential distributions, which follow from this observation.