# Transformations of stochastic matrix that preserve equilibrium

I have a stochastic (Markov) matrix $$W$$. I would like to modify it, such that $$W_{i,i}$$ increases for all $$i$$ (and thus other elements decrease). However, I don't want to change the equilibrium distribution of $$W$$, ie its leading eigenvector. Are there classes of transform that accomplish this?

For any $$0 \leq t < 1$$, the matrix $$(1 - t)W + tI$$ is a "lazier" version of your Markov chain that has the same equilibrium distribution.