  We present a coarse-graining (or model order reduction) procedure for
stochastic matrices by clustering. The method is consistent with the natural
structure of Markov theory, preserving positivity and mass, and does not rely
on any tools from Hilbert space theory. The reconstruction is provided by a
generalized Penrose-Moore inverse of the coarse-graining operator incorporating
the inhomogeneous invariant measure of the Markov matrix. As we show, the
method provides coarse-graining and reconstruction also on the level of tensor
spaces, which is consistent with the notion of an incidence matrix and quotient
graphs, and, moreover, allows to coarse-grain and reconstruct fluxes.
Furthermore, we investigate the connection with functional inequalities and
Poincar\'e-type constants.
