Unitary group $\U_n(\CC)$

Let \(n\geq 1\), and for \(A\in\Mat_{n\times n}(\CC)\) let \(A^\dagger\) be its conjugate transpose. The unitary group is \[ \U_n(\CC) \coloneqq \theset{ A\in\Mat_{n\times n}(\CC) \st A^\dagger A = AA^\dagger = I_n }. \]