<s>[INST] What kind of matrix is this and why does this happen? So I was studying Markov chains and I came across this matrix  \begin{align*}P=\left( \begin{array}{ccccc}
        0 & \frac{1}{4} & \frac{3}{4} & 0 & 0\\
        \frac{1}{4} & 0 & 0 & \frac{1}{4} & \frac{1}{2}\\
        \frac{1}{2} & 0 & 0 & \frac{1}{4}& \frac{1}{4}\\
        0 & \frac{1}{4} & \frac{3}{4} & 0& 0\\
        0 & \frac{1}{4} & \frac{3}{4} & 0& 0\\
    \end{array} \right).\end{align*}
I noticed (by brute force) that \begin{align*}P^2=\left( \begin{array}{ccccc}
        \frac{7}{16} & 0 & 0 & \frac{1}{4} & \frac{5}{16}\\
        0 & \frac{1}{4} & \frac{3}{4} & 0& 0\\
        0 & \frac{1}{4} & \frac{3}{4} & 0& 0\\
        \frac{3}{8} & 0 & 0 & \frac{1}{4}& \frac{1}{2}\\
        \frac{7}{16} & 0 & 0 & \frac{1}{4} & \frac{5}{16}\\
    \end{array} \right),\end{align*}
and \begin{align*}P^3=\left( \begin{array}{ccccc}
        0 & \frac{1}{4} & \frac{3}{4} & 0& 0\\
        \frac{7}{16} & 0 & 0 & \frac{1}{4} & \frac{5}{16}\\
        \frac{7}{16} & 0 & 0 & \frac{1}{4} & \frac{5}{16}\\
        0 & \frac{1}{4} & \frac{3}{4} & 0& 0\\
        0 & \frac{1}{4} & \frac{3}{4} & 0& 0\\
    \end{array} \right).\end{align*}
In fact; using a computer I found that every even power takes the form of the $P^2$ matrix and every odd power takes the form of the $P^3$ matrix.
I just wanted to know why that oscillation occurs? Is there a special name for the kind of matrix that $P$ is for it to exhibit that kind of behaviour?
 [/INST] This is a periodic Markov chain (with period $2$).  Otherwise, there's not much that's terribly unusual about it.

User "Iwillnotexist Idonotexist" raised an important point in the comments:

Well, something that can be noted for periodic Markov chains is that
  by definition they cannot be ergodic, another very important property
  of MCs that you may encounter soon. Roughly speaking, an ergodic MC
  that runs long enough "forgets" everything about its initial state. If
  the MC is periodic, then clearly you must remember some information
  about the contents of the initial state, because you're stuck in a
  loop of states that you keep on coming back to and aren't forgetting.

</s>