Finding the eigenvectors of a reflection matrix How could I find the eigenvectors of the following matrix? 
$$\left(\begin{matrix} \cos(\theta) & \sin(\theta)\\ \sin(\theta)& -\cos(\theta)\end{matrix}\right)$$
I found the eigenvalues $1$ and $-1$ but I'm struggling with calculating the eigenvectors.

Let $$A = \left(\begin{matrix}\cos(\theta) & \sin(\theta)\\ \sin(\theta)& -\cos(\theta)\end{matrix}\right).$$
$\lambda = +1,-1$ as you have already mentioned.
So eigenvectors corresponding to $\lambda = 1$ is got by solving for $x$ in $$Ax = x.$$
So do the row echelon conversion (or any other method to solve the simultaneous equation) and we get, $$ x = \begin{bmatrix}
           \frac{-\sin(\theta)}{\cos(\theta)-1} \cdot c \\
           c
         \end{bmatrix},$$ where $c \in \mathbb{R}$.
Similarly one gets eigenvectors span for $\lambda = -1$.
