Two dimensional representation of $D_5$

Question

I am trying to figure out the $2$-dim representation of $D_5$. Consider the action of $D_5$ on $\mathbb{R}^2$ by rotation and reflection. Then we can define a $2$-dim representation $\rho:D_5\to GL_2(\mathbb{R})$ s.t. $x\mapsto \begin{pmatrix}\cos{\frac{2\pi}{5}}&-\sin{\frac{2\pi}{5}}\\\sin{\frac{2\pi}{5}}&\cos{\frac{2\pi}{5}}\end{pmatrix}$, $y\mapsto \begin{pmatrix}0&1\\1&0\end{pmatrix}$. Then I can check the irreducibility by checking $\langle\chi,\chi\rangle=1$. Similarly, we have another $2$-dim irreducible representation by mapping $x$ to the rotation matrix by degree $4\pi/5$. However, my question is that how to check whether it's irreducible without using character theory. Geometrically, in what degree of rotation that $x$ maps to the corresponding $2$-dim representation will be irreducible?

Answer 1

If it was reducible, both matrices would have a common eigenvector. Calculate eigenvectors, show that there is no common one.