Reposting from [math.sx][1] 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 unique **left** eigenvector called perron vector $x$ corresponding to the largest eigenvalue and also it sums to 1. Call it as $x$. Let $D$ be a diagonal matrix such that $B=DA$ is a stochastic matrix with rows summing to 1. Call the perron vector of $B$ as $y$. How are $x$ and $y$ related?


  [1]: http://math.stackexchange.com/questions/1468447/how-does-scaling-rows-to-sum-to-1-of-a-positive-matrix-change-the-perron-vector