Editing Normal operator (section)

{{Short description|(on a complex Hilbert space) continuous linear operator}}
{{Refimprove|date=June 2011}}
In [[mathematics]], especially [[functional analysis]], 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>N^{\ast}N = NN^{\ast}</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. 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., self-adjoint operators): <math>N^{\ast} = N</math>
* [[skew-Hermitian]] operators: <math>N^{\ast} = -N</math>
* [[positive operator]]s: <math>N = M^{\ast}M</math> for some <math>M</math> (so ''N'' is self-adjoint).

A [[normal matrix]] is the matrix expression of a normal operator on the Hilbert space <math>\mathbb{C}^{n}</math>.