Editing Operator theory (section)

====Normal operators====
{{main article|Normal operator}}

A '''normal operator''' on a [[complex number|complex]] [[Hilbert space]] <math>H</math> is a [[continuous function (topology)|continuous]] [[linear operator]] <math>N\colon H \rightarrow H</math> that [[commutator|commutes]] with its [[hermitian adjoint]] <math>N^{\ast}</math>, that is: <math>NN^{\ast} = N^{\ast}N</math>.<ref>{{citation
 | last1 = Hoffman | first1 = Kenneth
 | last2 = Kunze | first2 = Ray | author2-link = Ray Kunze
 | edition = 2nd
 | location = Englewood Cliffs, N.J.
 | mr = 0276251
 | page = 312
 | publisher = Prentice-Hall, Inc.
 | title = Linear algebra
 | year = 1971}}</ref>

Normal operators are important because the [[spectral theorem]] holds for them. Today, the class of normal operators is well understood. Examples of normal operators are
* [[unitary operator]]s: <math>U^{\ast} = U^{-1}</math>
* [[Hermitian operator]]s (i.e., selfadjoint operators): <math>N^{\ast} = N</math>; (also, anti-selfadjoint operators: <math>N^{\ast} = -N</math>)
* [[positive operator]]s: <math>N = M^{\ast}M</math>, where <math>M</math> is any operator
* [[Normal matrix|normal matrices]] can be seen as normal operators if one takes the Hilbert space to be <math>\mathbb{C}^{n}</math>.

The spectral theorem extends to a more general class of matrices. Let <math>A</math> be an operator on a finite-dimensional [[inner product space]]. <math>A</math> is said to be [[normal matrix|normal]]  if <math>A^{\ast}A = AA^{\ast}</math>. One can show that <math>A</math> is normal if and only if it is unitarily diagonalizable: By the [[Schur decomposition]], we have <math>A = UTU^{\ast}</math>, where <math>U</math> is unitary and <math>T</math> [[upper triangular]]. Since <math>A</math> is normal, <math>T^{\ast}T = TT^{\ast}</math>. Therefore, <math>T</math> must be diagonal since normal upper triangular matrices are diagonal. The converse is obvious.

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