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Consider two irreducible finite state Markov chains with transition matrices $A,B\in\mathbb{R}^{n\times n}$.

Let $x$ and $y$ be the unique stationary distributions of $A$ and $B$, respectively.

Now consider the set of all stochastic matrices $P$ formed by taking the convex combination of $A$ and $B$, that is, $$P=\{M\in\mathbb{R}^{n \times n}\mid M=\lambda A+(1-\lambda)B, \lambda\in[0,1]\}$$

Is it possible to characterize the set of stationary distributions that arise from all matrices in the set $P$ in terms of (some properties of) $A$ and $B$?

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  • $\begingroup$ @SteveHuntsman Thanks Steve! Can you provide some intuition for the statement "if $xB$ and $yA$ are both in the span of $x$ and $y$..." The stationary distributions of $A$ and $B$ are $x$ and $y$, i.e., $xA=x$ and $yB=y$, I'm not quite seeing why $xB$ and $yA$ (note the switch) being in $S=\text{span}(\{x,y\})$ allows us to say that all stationary distributions of the matrices in $P$ are in $S$ as well. $\endgroup$
    – jonem
    Commented Mar 12, 2021 at 20:39
  • $\begingroup$ Sorry, my mistake. Retaining the first half of my original comment... Note that if 𝐵 is a convex combination of 𝐴 and the identity then the problem is trivial. $\endgroup$
    – Steve Huntsman
    Commented Mar 13, 2021 at 0:56

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