Suppose we have a stochastic matrix $P$ for a Markov chain, and we can compute a low rank approximation of $P$, say $P_k$, or we can find the nonnegative matrix factorization of $P$, i.e., $P=AW$ where $A$ (resp $W$) is a $n\times k$ (resp. $k\times n$) matrix.
What are the possible applications of these low-rank "techniques"? In particular, can I come up with a smaller Markov chain which "approximates" the original one? or can I use them to facilitate certain computation (e.g., to find the stationary distribution)?