On the Products of Stochastic and Diagonal Matrices

Consider a stochastic matrix $P$ and diagonal matrix $D\mathrm{.}$ In this work, we introduce Tilted matrices. A Tilted matrix is the product $D^{\mathrm{\prime }}PD$, where $D^{\mathrm{\prime }}$ is a diagonal normalization that makes the product stochastic. We then provide several results on products of Tilted matrices, which can be desirable for analyses of Markov Decision Processes. Lastly, we obtain a convergence rate result for the product of Tilted reversible matrices.