Quote by chiro
The determinant of a rotation matrix should always be 1 (since it preserves length) so there should always be eigenvalues and eigenvectors that can be calculated given a rotation matrix.

the characteristic polynomial of:
[tex]\begin{bmatrix}\cos\theta&\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}[/tex]
is:
[tex]x^2  (2\cos\theta)x + 1[/tex]
which has real solutions only when:
[tex]4\cos^2\theta  4 \geq 0 \implies \cos\theta = \pm 1[/tex]
for angles that aren't integer multiples of pi, this will lead to complex eigenvalues.