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P: 906
Rotation matrix vs regular matrix

 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:

$$\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}$$

is:

$$x^2 - (2\cos\theta)x + 1$$

which has real solutions only when:

$$4\cos^2\theta - 4 \geq 0 \implies \cos\theta = \pm 1$$

for angles that aren't integer multiples of pi, this will lead to complex eigenvalues.