Column stochastic matrix/0110/Periodic/Example

The column stochastic ${\displaystyle {}2\times 2}$-matrix

${\displaystyle {\begin{pmatrix}0&1\\1&0\end{pmatrix}}}$

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