A necessary condition for weak lumpability in finite Markov processes

Under certain conditions, the state space of a homogeneous Markov process can be partitioned to construct an aggregated Markovian process. However, the verification of these conditions requires expensive computations. This note exposes a necessary condition for having a Markovian aggregated process. This condition is based on properties of the eigenvalues of certain submatrices of the transition rate matrix of the original Markov process.