Editing Householder transformation (section)

===Operator and transformation===

The '''Householder [[Operator (mathematics)|operator]]'''<ref>{{harvnb|Roman|2008|loc=p. 243-244}}</ref> may be defined over any finite-dimensional [[inner product space]] <math> V</math> with [[inner product]] <math> \langle \cdot, \cdot \rangle </math> and [[unit vector]] <math> u\in V</math> as
:<math> H_u(x) := x - 2\,\langle x,u \rangle\,u\,.</math><ref>{{cite book|title=Methods of Applied Mathematics for Engineers and Scientist|date=28 June 2013 |publisher=Cambridge University Press|isbn=9781107244467|pages=Section E.4.11|url=https://books.google.com/books?id=nQIlAAAAQBAJ}}</ref>

It is also common to choose a non-unit vector <math>q \in V</math>, and normalize it directly in the Householder operator's expression:<ref>{{harvnb|Roman|2008|loc=p. 244}}</ref>
:<math>H_q \left ( x \right ) = x - 2\, \frac{\langle x, q \rangle}{\langle q, q \rangle}\, q \,.</math>

Such an operator is [[Linear operator|linear]] and [[self-adjoint]].

If <math>V=\mathbb{C}^n</math>, note that the reflection hyperplane can be defined by its ''normal vector'', a [[unit vector]] <math display="inline">\vec v\in V</math> (a vector with length <math display="inline">1</math>) that is [[orthogonal]] to the hyperplane. The reflection of a [[Point (geometry)|point]] <math display="inline">x</math> about this hyperplane is the '''Householder [[linear transformation|transformation]]''':

: <math>\vec x - 2\langle \vec x, \vec v\rangle \vec v = \vec x - 2\vec v\left(\vec v^* \vec x\right), </math>

where <math>\vec x</math> is the vector from the origin to the point <math>x</math>, and <math display="inline">\vec v^*</math> is the [[conjugate transpose]] of <math display="inline">\vec v</math>.

[[File:Householdertransformation.png|thumb|The Householder transformation acting as a reflection of <math>x</math> about the hyperplane defined by <math>v</math>.]]