Column stochastic matrix/2x2/Example

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

$\left(\begin{array}{cc}{p}_1& p_2\\ 1-p_1& 1-p_2\end{array}\right)$

wit

$0\le p_1,p_2\le 1.$

The characteristic polynomial is

$\begin{array}{cc}(X-p_1)(X-1+p_2)-(1-p_1)p_2& =X^2+(p_2-p_1-1)X+p_1(1-p_2)-p_2(1-p_1)\\ & =X^2+(p_2-p_1-1)X+p_1-p_2\\ & =(X-1)(X+p_2-p_1).\end{array}$

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 $\left(\begin{array}{c}\frac{p_2}{p_2-p_1+1}\\ \frac{1-p_1}{p_2-p_1+1}\end{array}\right)$, because of

$\begin{array}{cc}\left(\begin{array}{cc}{p}_1& p_2\\ 1-p_1& 1-p_2\end{array}\right)\left(\begin{array}{c}\frac{p_2}{p_2-p_1+1}\\ \frac{1-p_1}{p_2-p_1+1}\end{array}\right)& =\left(\begin{array}{c}{p}_1\frac{p_2}{p_2-p_1+1}+p_2\frac{1-p_1}{p_2-p_1+1}\\ (1-p_1)\frac{p_2}{p_2-p_1+1}+(1-p_2)\frac{1-p_1}{p_2-p_1+1}\end{array}\right)\\ & =\left(\begin{array}{c}\frac{p_1{p}_2+p_2(1-p_1)}{p_2-p_1+1}\\ \frac{(1-p_1)p_2+(1-p_2)(1-p_1)}{p_2-p_1+1}\end{array}\right)\\ & =\left(\begin{array}{c}\frac{p_2}{p_2-p_1+1}\\ \frac{1-p_1}{p_2-p_1+1}\end{array}\right).\end{array}$