Normal matrices

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A complex square matrix A is a normal matrix if

A^\dagger A=AA^\dagger ,

where A^\dagger is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose: [A,A^\dagger]=0.

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

A = U \Lambda U^\dagger,

where

 \Lambda = \mathrm{diag} (\lambda_1, \lambda_2, \dots)
 U^\dagger U = U U^\dagger= I.

The entries \lambda_i of the diagonal matrix \Lambda are the eigenvalues of A, and the columns of U are the eigenvectors of A. The matching eigenvalues in \Lambda must be ordered as the eigenvectors are ordered as columns of U.

References