How would you diagonalize:
\left(
\begin{array}{cc}
0 & -1\\
1 & 0
\end{array}
\right)
over the reals? The problem does not specify that V is a complex space.
Hm...can I bump this? I'm struggling with this same problem. First...this is a normal operator in an arbitrary inner product space, so I don't see how we know there is an orthonormal basis of eigenvectors. Also, I have a hunch that there is a less mechanical way to show this...