What can one say about the set of all $n$-dimensional square matrices $A \in \text{GL}_n(\mathbb{C})$ that have an inverse with entries out of $\mathbb{C}$ with the properties:
- unitary $:\Leftrightarrow A^*= A^{-1}$
- hermitian $:\Leftrightarrow A^* = A$
where $A^*$ is the conjugate transpose of $A$.
What obviously follows is $$A^{-1} = A$$ The most simple matrix that is in this set is the identity matrix. Are there others? How do they look like?