How to find solutions for linear recurrences using eigenvalues Use eigenvalues to solve the system of linear recurrences
$$y_{n+1} = 2y_n + 10z_n\\
z_{n+1} = 2y_n + 3z_n$$
where $y_0 = 0$ and $z_0 = 1$.
I have absolutely no idea where to begin. I understand linear recurrences, but I'm struggling with eigenvalues.

Set $x_n=[y_n z_n]^T$, and your system becomes $x_{n+1}=\left[\begin{smallmatrix}2&10\\2&3\end{smallmatrix}\right]x_n$.  Iteration becomes matrix exponentiation.  If your eigenvalues are less than 1 in absolute value, the matrix approaches 0.  If an eigenvalue is bigger than 1 in absolute value, you get divergence.  It's a rich subject, read about it on Wikipedia.
