TITLE: Eigenvalues of a rotation
QUESTION [5 upvotes]: How do I show that the rotation by a non zero angle $\theta$ in $\mathbb{R}^2 $ does not have any real eigenvalues. I know the matrix of a rotation but I don't how to show the above proposition. 
Thank you

REPLY [6 votes]: Let's do't without calculating the characteristic polynomial!
Remember that a putative eigenvector cannot be the zero vector, and that an eigenvector belonging to a real eigenvalue can be scaled to have real components. 
But if you rotate a vector in the real plane by an angle that is not an integer multiple of 180 degrees, the rotated version will not be parallel to the unrotated version. Therefore no such eigenvector exists (unless $\theta=n\cdot\pi$).