Discrete Markov Chains - PCCA+ and TPT

There are infinitely many transformations A of the eigenvectors resulting in a soft membership matrix χ satisfying the positivity and partition of unity constraints. Consequently, we have to determine the transformation A that satisfies some optimality condition. The starting point is the question whether it is possible to define some simplified dynamics on the coarse grained state space. Can we replace the original transition matrix T by a smaller transition matrix Tc ∈ Rnc×nc that propagates probability densities on the space of macrostates in a correct way? First of all, we note that densities on the original state space pf ∈ R can be restricted to a density pc ∈ Rc on the coarse state space by the restriction operator: