Let $A=\begin{pmatrix}1 & 1/2 \\ -1/2 & 0 \end{pmatrix}$. Define $(x_n)$ recursively by $$ x_{n+1}=A x_n $$ for some starting vector $x_0\neq 0\in\mathbb R^2$. Let $\vert\cdot\vert$ be the Euclidean norm on $\mathbb R^2$. I want to show that this iteration converges linearly, which means that $$ \lim_n \frac{\vert x_{n+1}\vert}{\vert x_n\vert}\neq 0. $$
Some observations:
Convergence is clear, since $1/2$ is the only eigenvalue. So let's move on to the showing that the limit exists and is non-zero. Numerically, it seems that the limit is equal to $1/2$, which is the spectral radius of $A$. If $x_0$ is an eigenvector of $A$, then it's clear from $x_{n+1}=Ax_n=1/2 x_n$ that the above quotient has limit $1/2$. However, how can I argue for $x_0$ which is not an eigenvector?
Maybe we can use that $$ \vert x_{n+1}\vert\leq \vert A\vert \vert x_n\vert, $$ where $\vert A\vert$ is the $2$-norm of $A$. Or maybe the Jordan-Canonical form is useful?