$$A=\begin{pmatrix} 2 & 0 &-1\\ 0 & 4 & 0 \\ -1 & 0 & 2 \end{pmatrix} $$ and let $\{y_n\}$ be three-dimensional number vector $$ y_1=\begin{pmatrix} 2 \\ 0 \\ 0 \end{pmatrix},y_{n+1}=Ay_n $$ then, what is the $$\lim_{n \to \infty} \frac{1}{n}\log| |y_n||\quad?$$
My attempt:
Since $$P=\begin{pmatrix} 1 & 1 &0\\ 0 & 0 & 1 \\ 1 & -1 & 0 \end{pmatrix} $$ $$P^{-1}AP=\begin{pmatrix} 1 & 0 &0\\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix} $$ But I have no idea what should be the next step. Can somone help me to solve ? Thank you in advance.
$$y_1=\begin{pmatrix}2 \\0 \\0 \end{pmatrix}=\begin{pmatrix}1 \\0 \\1 \end{pmatrix}+\begin{pmatrix}1 \\0 \\-1 \end{pmatrix}$$ $$y_2=Ay_1=A\begin{pmatrix}1 \\0 \\1 \end{pmatrix}+A\begin{pmatrix}1 \\0 \\-1 \end{pmatrix}=1\begin{pmatrix}1 \\0 \\1 \end{pmatrix}+3\begin{pmatrix}1 \\0 \\-1 \end{pmatrix}$$ $$y_{n+1}=1^n\begin{pmatrix}1 \\0 \\1 \end{pmatrix}+3^n\begin{pmatrix}1 \\0 \\-1 \end{pmatrix}$$ Easy to see that $$\frac{1}{n}\log| |y_{n}||\rightarrow \log3$$