I already asked this question in StackExchange, but found little attention. So I’m just going to copy-paste my original question here.

Let $ P$ be a stochastic matrix (of an irreducible Markov Chain) with stationary distribution $ \pi^T$ (i.e. $ \pi^T P = \pi^T$ ) and let further $ E$ be the matrix of all $ 1$ ‘s.

Given an $ \alpha \in [0,1]$ , is it possible to find an expression for the stationary distribution of $ $ \alpha P + \frac{(1-\alpha)}{n}E,$ $ depending on $ \pi$ and $ \frac{1}{n}\mathbb{1}$ , where $ \mathbb{1}$ is the vector of all $ 1$ ‘s?

More generally; given two transition matrices of irreducible Markov Chains $ P_1$ and $ P_2$ with stationary distributions $ \pi_1^T$ and $ \pi_2^T$ , respectively. Can one find a general formula to calculate the stationary distribution of $ $ \alpha P_1 + (1-\alpha)P_2 \quad,$ $ for $ \alpha \in [0,1]$ ?