There is a difference equation of Markov process.
$y_{1}=0.9y_{0}+0.2z_{0}$
$z_{1}=0.1y_{0}+0.8z_{0}$
Let matrix A =$\begin{pmatrix} 0.9 & 0.2 \\ 0.1 & 0.8 \end{pmatrix}$
Then from det(A-$\lambda$I)=0
$\lambda_{1}=1$ and $\lambda_{2}=0.7$
A=$S\lambda S^{-1}$=$\begin{pmatrix} \frac{2}{3} & \frac{1}{3} \\ \frac{1}{3} & -\frac{1}{3} \end{pmatrix}$ $\begin{pmatrix} 1 & 0 \\ 0 & 0.7 \end{pmatrix}$ $\begin{pmatrix} 1 & 1 \\ 1 & -2 \end{pmatrix}$
Why is this? I think the eigenvectors are $x_{1}$=(2 1) and $x_{2}$=(1 -1) but can't understand why the eigenvector matrix S is formed like that.