Consider a Matrix $M$ with non-negative entries and it's power $M^n$. Assume we know every entry of $M^n$ denoted by $m_{ij;n}$. Now we multiply the $i$-th row of $M$ by a positive factor $\alpha$ which yields $M'$. What can we say about the relation between $m_{ij;n}$ and $m'_{ij;n}$?
$m'_{ij;n} \geq \alpha m_{ij;n}$ should hold, but can we say more than this?
Does the following work:
(1) $m'_{ii;n} = \sum_k m'_{ik;1} m'_{ki;n-1} \geq m'_{ii;1} m'_{ii;n-1} \geq \cdots \geq m'^n_{ii;1} \geq \alpha^n m^n_{ii;1}$
I also thought along those lines: multiplying a row can be seen as multiplying by an appropriate diagonal matrix $D$. Hence the question becomes: How do entries of $(DM)^n$ correspond to entries of $M^n$?