Spectral theorem for unitary matrices

Statement
This has the following equivalent forms. Let $$U(n,\mathbb{C})$$ denote the unitary group: the group of $$n \times n$$ unitary matrices over complex numbers. Then:


 * Any element of $$U(n,\mathbb{C})$$ is conjugate, in $$U(n,\mathbb{C})$$, to a diagonal unitary matrix
 * The subgroup of $$U(n,\mathbb{C})$$ comprising diagonal unitary matrices, is conjugate-dense in $$U(n,\mathbb{C})$$

The result follows from the spectral theorem for normal matrices, which also implies the spectral theorem for Hermitian matrices.