On finite interweaving relations

A faithful bi-interweaving relation is a Markovian similarity-type relation between two Markov chains, strengthening the Markovian intertwining relation and introducing warming-up times after which the time-marginal distributions of the chains can be tightly compared (for any initial distributions). For irreducible transition kernels on the same finite state space, these relations are shown to be equivalent to the generalised isospectrality relation, but this is no longer true for nontransient transition kernels, contrary to the faithful bi-intertwining relations. Some bounds are deduced on corresponding warming-up times, when the eigenvalues are furthermore assumed to be real (but still allowing for Jordan blocks). When the eigenvalues are non-negative, the same approach enables us to construct strong stationary times for irreducible Markov chains through interweaving relations with model absorbed Markov chains, thus extending a result due to Matthews in the reversible situation.