# Spectral theorem for unitary matrices

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This article gives the statement, and proof, of a particular subgroup in a group being conjugate-dense: in other words, every element of the group is conjugate to some element of the subgroup

## 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.