Reachability of consensus and synchronizing automata

We consider the problem of determining the existence of a sequence of matrices driving a discrete-time multi-agent consensus system to consensus. We transform this problem into the problem of the existence of a product of the (stochastic) transition matrices that has a positive column. This allows us to make use of results from automata theory to sets of stochastic matrices. Our main result is a polynomial-time algorithm to decide the existence of a sequence of matrices achieving consensus.