How to prove $O(n,C) $ is not compact

Question

How does one prove that $O(n,C) $ is not compact?

I am guessing it can be done by showing it is not bounded.

Answer 1

Try this :

$$ \begin{bmatrix} r_n & i\sqrt {r_n^2-1} \\ -i\sqrt {r_n^2-1} & r_n \\ \end{bmatrix} $$

where $r_n\in \mathbb C$

Answer 2

Try this :

$$ \begin{bmatrix} \cos z& -\sin z \\ \sin z & \cos z \\ \end{bmatrix} $$

where $z\in \mathbb C$