Normal matrices

A complex square matrix A is a normal matrix if

$A^{†}A=A{A}^{†},$

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

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 $\mathrm{\Lambda }$ and a unitary matrix $U$ by the formula

$A=U\mathrm{\Lambda }{U}^{†},$

where

$\mathrm{\Lambda }=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}({\lambda }_1,{\lambda }_2,\dots )$

$U^{†}U=U{U}^{†}=I.$

The entries ${\lambda }_i$ of the diagonal matrix $\mathrm{\Lambda }$ are the eigenvalues of $A$, and the columns of $U$ are the eigenvectors of $A$. The matching eigenvalues in $\mathrm{\Lambda }$ must be ordered as the eigenvectors are ordered as columns of $U$.

References

• Normal matrix entry in Wikipedia