I have to following problem to solve:
Show that every matrix A $∈\ O_2$ \ $SO_2$ is diagonalizable, and determine one for A similar diagonal matrix.
What I already know: $SO_2=\{ \begin{pmatrix} \cos(\alpha) & -sin(\alpha) \\ \ sin(\alpha) & \ cos(\alpha) \end{pmatrix} \mid \alpha\in [0,2\pi) \} $
And $O_2 = $$SO_2 \cup SO_2$$ \cdot \begin{pmatrix} 1 & 0 \\ \ 0 & \ -1 \end{pmatrix} $
I am stuck on the problem, and don't know how to do it. I already looked into my skript a couple of times now..
Every matrix in $O \setminus SO$ can be written in the form $$ \pmatrix{\cos \alpha & -\sin \alpha\\ \sin \alpha & \cos \alpha} \pmatrix{1&0\\0&-1} = \pmatrix{\cos \alpha & \sin \alpha \\ \sin \alpha & - \cos \alpha} $$ Verify that for every $\alpha$, the matrix on the right has eigenvalues $1,-1$. The rest follows.