The coupled recurrence system $x_n = 2x_{n-1} + y_{n-1}$ $y_n = x_{n-1} + 2y_{n-1}$ can be written as $\begin{pmatrix} x_n \\ y_n \end{pmatrix} = M \begin{pmatrix} x_{n-1} \\ y_{n-1} \end{pmatrix}$ where $M = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$.

What are the eigenvalues (characteristic roots) of $M$?