Given a stochastic (all rows/columns, depending on convention, sum to 1; not generally symmetric) matrix $P$ and positive diagonal matrix $D$, is there a name for the matrices of the form $PD$? This matrix product is important in the use of Markov Additive Processes, where $D$ represents an exponential tilting factor, wrt dynamics $P$.
Of course, any non-negative matrix can be factorized this way. However, I am moreover interested in what happens if we change $D\to D'$ some other diagonal matrix. Some examples are if we perform a Hadamard/Schur/elementwise exponentiation $D \to D^{(\gamma)}$, if we swap rows and columns with a permutation matrix $S$: $D \to S^{-1} D S$.
The idea is that I'd like to use information about $PD$ to inform myself (in terms of approximations or inequalities) of the properties (especially Perron-Frobenius eigenvalue and eigenvectors) of $PD'$, where $D'$ is some transformed, positive diagonal matrix, with examples I've mentioned above. So: Are there names of these matrices and transformations? Are there results known or general literature I can read on?
Any thoughts or relevant literature for this area of research would be much appreciated.