Reposting from math.sx due to lack of response. Let $A$ be a $N\times N$ positive matrix such that $A_{ij}>0$. By Perron-Frobenius theorem, there is a unique positive left eigenvector called Perron vector $x$ corresponding to the largest eigenvalue and also it sums to 1. Call it $x$. Let $D$ be a diagonal matrix such that $B=DA$ is a stochastic matrix with rows summing to 1. Call $y$ the Perron vector of $B$. How are $x$ and $y$ related?