Linear Algebra: Geometric means What is the geometric means of  
$$M=\begin{pmatrix}\cos \theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$$
I would like to show that its eigenvector is not real.

As already noted, the matrix represents a rotation about the origin. Let's also assume $\theta\neq \pi n$ for any integer $n$, otherwise your matrix is $\pm$identity and does have real eigenvectors.
Anyways, the intuition is that eigenvectors of $A$ are non-zero vectors that are scaled by $A$, i.e., end up in the same direction but with a possibly different length. Can this happen in your case?
