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.