For a matrix $\mathbf{A} \in \mathbb{R}^{2\times2}$, what can one say about its eigenvalues $\gamma_1, \gamma_2 \in \mathbb{C}$, if:
$$\mathbf{S}_0 \in \mathbb{R}^{2}$$
$$\mathbf{S}_{n+1} = \mathbf{A}\mathbf{S}_{n}$$
and for all $\mathbf{S}_i$ generated this way, the following holds: $\|\mathbf{S}_{i}\| = \|\mathbf{S}_{0}\|$.
That is, the Euclidean norm of $\mathbf{S}_0$ is preserved even under repeated action of $\mathbf{A}$.
Alternatively, what must the eigenvalues of $\mathbf{A}$ look like if the above holds true?
Well, the condition must be true for any eigenvector $v$ of eigenvalue $\gamma$, hence
$$|\gamma| . \| v \| = \| Av \| = \| v \| \ \ \Rightarrow |\gamma| = 1.$$