Koszul algebra: Difference between revisions
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In [[Linear algebra]], define the '''Householder operator''' as follows. | |||
Let <math> V\, </math> be a finite dimensional [[inner product space]] with [[unit vector]] <math> u\in V</math> Then, the Householder operator is an [[Operator (mathematics)|operator]] <math> H_u : V \to V\,</math> defined by | |||
:<math> H_u(x) = x - 2\langle x,u \rangle u\,</math> | |||
where <math> \langle \cdot, \cdot \rangle </math> is the [[inner product]] over <math>V\,</math>. This operator reflects the vector <math>x</math> across a plane given by the normal vector <math>u</math>.<ref>{{cite book|title=Methods of Applied Mathematics for Engineers and Scientist|publisher=Cambridge University Press|isbn=9781107244467|pages=Section E.4.11|url=http://books.google.com/books?id=nQIlAAAAQBAJ}}</ref> | |||
Over a [[real vector space]], the Householder operator is also known as the [[Householder transformation]]. | |||
The Householder operator has numerous properties such as linearity, being [[self-adjoint]], and is a [[Unitary operator|unitary]] or [[orthogonal]] operator on V. | |||
==References== | |||
{{reflist}} | |||
[[Category:Numerical linear algebra]] | |||
{{Linear-algebra-stub}} | |||
Revision as of 04:08, 5 May 2013
In Linear algebra, define the Householder operator as follows.
Let be a finite dimensional inner product space with unit vector Then, the Householder operator is an operator defined by
where is the inner product over . This operator reflects the vector across a plane given by the normal vector .[1]
Over a real vector space, the Householder operator is also known as the Householder transformation.
The Householder operator has numerous properties such as linearity, being self-adjoint, and is a unitary or orthogonal operator on V.
References
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