 Solution

The upper-left block is a planar rotation matrix. Its real characteristic polynomial is $(\lambda -\cos \theta {)}^2+{\sin }^2\theta$, which is strictly positive for real $\lambda$ because the angle excludes integer multiples of $\pi$. Hence any real eigenvector has $x=y=0$. The remaining equation is $5z=\lambda z$, givingThere are infinitely many nonzero solutions, spanning one independent eigenvector direction.