Suppose we have $k$ square ($n$ by $n$) stochastic/probability matrices, $M_1\dots M_k$. So $\sum_j (M_k)_{ij} = 1 $ for all $i$. Suppose we have a probability vector $p$ with $k$ non-negative components, $\sum_j p_j = 1$. Define the new stochastic matrix $M=\sum_j p_jM_j$ (so a convex combination of the $M_j$ matrices).
If we know the eigenvectors of each $M_j$, what can we say about the eigenvectors of $M$?