(Important: THIS PROBLEM IS NOT DUPLICATED! Note that the case where just one row of $W$ is multiplied by constant $c$, can be handled by the Sherman-Morrison theorem, but the case where the whole matrix $W$ is multiplied by some constant $c$ does not solve that easy)
Assume $W$ is a non-negative $n\times n$ row stochastic matrix and $r<1$ is a real number. Let $$Q = \sum_{i=0}^{\infty} (rW)^i=[I_n-rW]^{-1}$$.
Now assume that the matrix $W$ is multiplied by a constant real number $c<1$. Let this new matrix $U$. Let $$P=\sum_{i=0}^{\infty} (rU)^i=[I-rU]^{-1}=[I-rcW]^{-1}$$ I want to know to know the relation between $Q$ and $P$ and if we can express entries of $P$ in terms of entries of $Q$. Or is it possible to have bound on $P$ in terms of $Q$?
Thanks.
This case is much simply than the one in your previous question. We have $I-P^{-1}=rcW=c(I-Q^{-1})$ and hence $P=\left(I-c(I-Q^{-1})\right)^{-1}$.