Column stochastic matrix/0110/Periodic/Example

The column stochastic $2\times 2$-matrix

$\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$

transforms the distribution $\left(\begin{array}{c}p\\ 1-p\end{array}\right)$ into the distribution $\left(\begin{array}{c}1-p\\ p\end{array}\right)$. The contribution $\left(\begin{array}{c}\frac{1}{2}\\ \frac{1}{2}\end{array}\right)$ is transformed to itself, that is, it is a stationary distribution. The distribution $\left(\begin{array}{c}1\\ 0\end{array}\right)$ is transformed into $\left(\begin{array}{c}0\\ 1\end{array}\right)$, and vice versa. It is a periodic distribution, the period length is $2$.