Column stochastic matrix/2x2/Example

A column stochastic ${}2\times 2$-matrix has the form

${\begin{pmatrix}p_{1}&p_{2}\\1-p_{1}&1-p_{2}\end{pmatrix}}$

wit

${}0\leq p_{1},p_{2}\leq 1\,.$

The characteristic polynomial is

${}{\begin{aligned}(X-p_{1})(X-1+p_{2})-(1-p_{1})p_{2}&=X^{2}+(p_{2}-p_{1}-1)X+p_{1}(1-p_{2})-p_{2}(1-p_{1})\\&=X^{2}+(p_{2}-p_{1}-1)X+p_{1}-p_{2}\\&=(X-1)(X+p_{2}-p_{1}).\,\end{aligned}}$

Eigenvalues are ${}1$ and ${}p_{1}-p_{2}$. A stationary distribution is (exclude the cases ${}p_{1}=1$ and ${}p_{2}=0$ for the following computation) given by ${}{\begin{pmatrix}{\frac {p_{2}}{p_{2}-p_{1}+1}}\\{\frac {1-p_{1}}{p_{2}-p_{1}+1}}\end{pmatrix}}$, because of

${}{\begin{aligned}{\begin{pmatrix}p_{1}&p_{2}\\1-p_{1}&1-p_{2}\end{pmatrix}}{\begin{pmatrix}{\frac {p_{2}}{p_{2}-p_{1}+1}}\\{\frac {1-p_{1}}{p_{2}-p_{1}+1}}\end{pmatrix}}&={\begin{pmatrix}p_{1}{\frac {p_{2}}{p_{2}-p_{1}+1}}+p_{2}{\frac {1-p_{1}}{p_{2}-p_{1}+1}}\\(1-p_{1}){\frac {p_{2}}{p_{2}-p_{1}+1}}+(1-p_{2}){\frac {1-p_{1}}{p_{2}-p_{1}+1}}\end{pmatrix}}\\&={\begin{pmatrix}{\frac {p_{1}p_{2}+p_{2}(1-p_{1})}{p_{2}-p_{1}+1}}\\{\frac {(1-p_{1})p_{2}+(1-p_{2})(1-p_{1})}{p_{2}-p_{1}+1}}\end{pmatrix}}\\&={\begin{pmatrix}{\frac {p_{2}}{p_{2}-p_{1}+1}}\\{\frac {1-p_{1}}{p_{2}-p_{1}+1}}\end{pmatrix}}.\end{aligned}}$