Editing Spectral theorem (section)

=== Normal matrices ===
{{main|Normal matrix}}
The spectral theorem extends to a more general class of matrices. Let {{math|''A''}} be an operator on a finite-dimensional inner product space. {{math|''A''}} is said to be [[normal matrix|normal]]  if {{math|1=''A''<sup>*</sup>''A'' = ''AA''<sup>*</sup>}}. 

One can show that {{math|''A''}} is normal if and only if it is unitarily diagonalizable using the [[Schur decomposition]]. That is, any matrix can be written as {{math|1=''A'' = ''UTU''<sup>*</sup>}}, where {{math|''U''}} is unitary and {{math|''T''}} is [[upper triangular]].
If {{math|''A''}} is normal, then one sees that {{math|1=''TT''<sup>*</sup> = ''T''<sup>*</sup>''T''}}. Therefore, {{math|''T''}} must be diagonal since a normal upper triangular matrix is diagonal (see [[normal matrix#Consequences|normal matrix]]). The converse is obvious.

In other words, {{math|''A''}} is normal if and only if there exists a [[unitary matrix]] {{math|''U''}} such that
<math display="block">A = U D U^*,</math>
where {{math|''D''}} is a [[diagonal matrix]]. Then, the entries of the diagonal of {{math|''D''}} are the [[eigenvalue]]s of {{math|''A''}}. The column vectors of {{math|''U''}} are the eigenvectors of {{math|''A''}} and they are orthonormal. Unlike the Hermitian case, the entries of {{math|''D''}} need not be real.