<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] The entries are all between 0 and 1 and in each row the entries add up to 1. They are called stochastic matrices. They have 1 as an eigen value with $(1,1,\ldots,1)^T$ as corresponding eigenvector. This property is inherited by all the powers of that matrix. 
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