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# Spaltenstochastisch/2x2/Beispiel

Eine spaltenstochastische ${\displaystyle {}2\times 2}$-Matrix hat die Form

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

mit

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

Eigenwerte sind also ${\displaystyle {}1}$ und ${\displaystyle {}p_{1}-p_{2}}$. Eine stationäre Verteilung ist (der Fall ${\displaystyle p_{1}=1}$ und ${\displaystyle p_{2}=0}$ ist für die folgende Rechnung auszuschließen) durch ${\displaystyle {}{\begin{pmatrix}{\frac {p_{2}}{p_{2}-p_{1}+1}}\\{\frac {1-p_{1}}{p_{2}-p_{1}+1}}\end{pmatrix}}}$ gegeben, es ist ja

{\displaystyle {}{\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}}}