I have the following problem for complex matrices: $G$ is a positive-definite, hermitian 2x2-matrix and $W$ an invertible 2x2 matrix. Moreover, $WGW^\dagger = G$ holds. Does this imply that $W$ is unitary (with respect to the standard scalar product)?
(Edit: The answer is NO. See the counter example below!) At first sight, it seemed obvious to me that the answer is YES. However, I still could not prove it. One should probably use the fact that $G = UDU^\dagger$ for unitary $U$ and a positive-definite diagonal matrix $D$. Furthermore, $\text{det}(WGW^\dagger) = \text{det}(W) \text{det}(G) \text{det}(W^\dagger) = \text{det}(G)$ and combined with $\text{det}(G) >0$ this yields $|\text{det}(G)| = 1$.
This is indication for the fact that $W$ could be unitary, but it's of course not enough.
Any help is welcome! Thanks a lot!
Cheers Qwerkopf
PS: What would the answer be in case of dxd-matrices?