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 $U$.