I came up with this question after reading the proof that a upper-triangular unitary matrix is diagonal. I was wondering if we can say anything more about the diagonal entries.
I came up with this proof:
Let $A \in \mathbb{C^{n x n}}$ be a unitary and diagonal matrix. Since $A$ is unitary,
$$A^* = A^{-1}$$
The diagonal entries of the inverse of a diagonal matrix are reciprocals. Equating their diagonal entries gives
$$\overline{a_{ii}}=\frac{1}{a_{ii}}$$
Since $a_{ii} \neq 0$
$$1 = a_{ii}\overline{a_{ii}}$$
Thus all diagonal entries of $A$ must satisfy the last equality.
This seems correct to me, but I can't find any reference that confirms it. Did I do anything wrong?