A Dual Eigenvector Condition for Strong Lumpability of Markov Chains

Necessary and sufficient conditions for identifying strong lumpability in Markov chains are presented. We show that the states in a lump necessarily correspond to identical elements in eigenvectors of the dual transition matrix. If there exist as many dual eigenvectors that respect the necessary condition as there are lumps in the aggregation, then the condition is also sufficient. The result is demonstrated with two simple examples.