In one of my lecture notes it is stated that one of the Eigenvalues (EV) of a stochastic Matrix can be calculated by $P_{11} - P_{21}$
I.e I have the following matrix : $$\begin{matrix} 0.1 & 0.9 \\ 0.5 & 0.5 \\ \end{matrix}$$ Obviously the EV are 1 and -0.4
Okay lets look at some more random Numbers : $$\begin{matrix} 0.42653 & 0.57347 \\ 0.2956 & 0.7044 \\ \end{matrix}$$ $P_{11} - P_{21}$ = 0.42653 - 0.2956 = 0.13093 which is its EV. The other is obviously 1
Is that always right for these kind of stochastic matrices?
When $P$ is a $2\times2$ row-stochastic matrix, one of its eigenvalues is $1$. Let the other eigenvalue be $\lambda$. Then $$ \lambda+1=\operatorname{tr}(P)=p_{11}+p_{22}=p_{11}+(1-p_{21}). $$ Therefore $\lambda=p_{11}-p_{21}$.
Alternatively, let $$ P=\pmatrix{p_{11}&p_{12}\\ p_{21}&p_{22}}=\pmatrix{a&1-a\\ b&1-b}. $$ Then $$ \pmatrix{1&-1}P=\pmatrix{1&-1}\pmatrix{a&1-a\\ b&1-b}=(a-b)\pmatrix{1&-1}. $$ Therefore $a-b=p_{11}-p_{21}=p_{22}-p_{12}$ is an eigenvalue of $P$.