Markov chain, multiplication of the matrix of transitions probabilities with itself

Question

Suppose that we have a Markov chain with the matrix of transition probabilities $p_{ij}(n)=P$. What tells us the square $P^2$ or higher order multiples of the matrix $P$ with itself; how it can be interpreted as another Markov Chain?

Answer 1

$P_2 = P^2$ is considering two jumps in the chain with matrix $P$ to be one jump in the chain with matrix $P_2$. If we ignore what happens every second frame.

The simplest example is probably:

$$P = \left[\begin{array}{cc}0&1\\1&0\end{array}\right], P_2 = P^2 = \left[\begin{array}{cc}1&0\\0&1\end{array}\right]$$

We simply become blind to the "flicker" of every second state. Examples we can draw parallells to (in some sense) include interlacing in old-school TV formats, or on a tic-toc clock we will only hear either the tics or the tocs.