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.

This matrix rotates the plane by $\theta$ degrees anti-clockwise. To find its eigen values, simply solve the equation
$$
det(A-\lambda I) = 0
$$
You will get
$$
(\cos(\theta) - \lambda)^2 +\sin^2(\theta) = 0
$$
$$
\Leftrightarrow 1 - 2\cos(\theta)\lambda + \lambda^2 = 0
$$
$$
\Leftrightarrow \lambda = \cos(\theta) \pm \sqrt{\cos^2(\theta) - 1}
$$
So your eigen values are $e^{\pm i\theta}$
