# Normal matrices

A complex square matrix A is a normal matrix if

where is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose: .

Normal matrices are precisely those to which the *spectral theorem* applies: a matrix is normal if and only if it can be represented by a diagonal matrix and a unitary matrix by the formula

where

The entries of the diagonal matrix are the eigenvalues of , and the columns of are the eigenvectors of . The matching eigenvalues in must be ordered as the eigenvectors are ordered as columns of .