Eigenvalues of a rotation 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 

The characteristical polynomial is 
$$x^2-2\cos(\theta) x+1$$
and $x^2\geq 0$ and $1>0$ and $|\cos(\theta)|\leq 1$  the polynomial can only have a  zero 
when $|\cos(\theta)|=1$.
As $x^2-2x+1=(x-1)^2$ and $x^2+2x+1=(x+1)^2$
