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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?

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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.

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  • $\begingroup$ Not exactly what I had in mind but that definitely works $\endgroup$ – mbarete May 20 at 16:32

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