# Inverse of Lumped Markov Process

Suppose we have an (ergodic, with dumping factor $\alpha$) Markov chain $P$ on a graph $G$, and we lump the transition matrix (it's a sort of reduction of the matrix using a equivalence relation of transition between states) obtaining $\bar{P}$.

We know that $P$ can be thought as the inverse of M-Matrix (specifically $(I-\alpha A)^{-1}$), and thus the matrix $P$ can have some interesting properties. I'm wondering if the new Markov process can be thought again an inverse of a M-Matrix, and thus have the same properties.