In Jalles2009pg5 they represented the determnistic sinusoidal cycle
$ \Psi_t = \alpha \cos(\lambda t) + \beta \sin(\lambda t),\quad t=1,2,\ldots$
via the recursion
$ \begin{bmatrix} \Psi_t \\ \Psi_t^* \end{bmatrix} = \begin{bmatrix} \cos(\lambda) & \sin(\lambda) \\ -\sin(\lambda) & \cos(\lambda) \end{bmatrix} \begin{bmatrix} \Psi_{t-1} \\ \Psi_{t-1}^* \end{bmatrix}$
with $\Psi_0 = \alpha$ and $\Psi_0^* = \beta$
I can algebraically derive that this works (the form at this Wikipedia link basically gets you there). But this still sort of seems like magic to me. I'm missing the bigger picture of why this works.
Question: Can anyone provide an intuitive explanation of why this works and, maybe, how this generalizes/simplifies?
