The Markov chainmcis irreducible if every state is reachable from every other state in at mostn- 1 steps, wherenis the number of states (mc.NumStates). This result is equivalent to (I+Z)^{n- 1}containing all positive elements.Iis then-by-nidentity matrix andZ_ij=I(P_ij> 0), for alli,j, which is the zero-pattern matrix of the transition matrixP(mc.P)[2]. To determine reducibility,isreduciblecomputes this matrix power.

By the Perron-Frobenius Theorem[2], irreducible Markov chains have unique stationary distributions. Unichains, which consist of a single recurrent class plus transient classes, also have unique stationary distributions (with zero probability mass in the transient classes). Reducible chains with multiple recurrent classes have stationary distributions that depend on the initial distribution.