Let $P$ be the transition probability matrix for a Markov chain, with each row summing to 1. Suppose P is 2x2.
Then, $\pi_k = (P^T)^k \pi_0$.
I want to show the speed of convergence of distribution to the stationary distribution $\pi_\infty$ is geometric with respect to the second largest eigenvalue.
If $P^T$ is diagonalizable, then I can start by writing $\pi_0$ as a linear combination of eigenvectors of $P^T$ so that:
Let $\pi_\infty$ be the stationary distribution.
$\pi_0 = c_1 \pi_\infty + c_2 \pi_2$
$\pi_k = c_1 \pi_\infty + c_2 \lambda^k \pi_2 $.
Q1: Is this correct so far? I am not so sure because $\pi_k \to c_1 \pi_\infty$ as $k \to \infty$ ? Shouldn't it be $\pi_k \to \pi_\infty$ instead?
Q2: How to deal with the case when $P^T$ is not diagonalizable?