# Spectral theorem for unitary matrices

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 $U(n,\mathbb{C})$