Taken from Groups and Symmetry by M.A. Armstrong, question 18.8:
Setup:
A 5x1 rectangular strip of paper is marked off on both sides into 5 unit squares. The two ends of the paper are then put together with a half twist, to create a Mobius band. Think of the Mobius band as the subset of $\mathbb C$ x $\mathbb C$: $\{(e^{2i\theta}, \lambda e^{i\theta}) | -\pi \le \theta \le \pi\}$.
The transformation which sends $(e^{2i\theta}, \lambda e^{i\theta})$ to $(e^{2i(\theta+\pi/5)}, \lambda e^{i(\theta+\pi/5)})$, and the transformation which sends $(e^{2i\theta}, \lambda e^{i\theta})$ to $(e^{-2i\theta}, \lambda e^{-i\theta})$ both send the marked Mobius band back to itself.
I'm trying to come up with matrices to generate these transformations. For the first one, I have:
$\begin{bmatrix} e^{i\frac{2\pi}{5}} & 0 \\ 0 & e^{i\frac{2\pi}{5}} \end{bmatrix}$, which I think works(?). But the second one has been more elusive... If I knew what $\theta$ of the input was in advance, I could just make a matrix of the form: $\begin{bmatrix} e^{-4i\theta} & 0 \\ 0 & e^{-2i\theta} \end{bmatrix}$, but since I want one matrix that does this for any input, I'm not sure what to do. Any help would be greatly appreciated!