In linear algebra, the Householder operator is defined as follows. Let
V
\langle ⋅ , ⋅ \rangle
u\inV
Hu:V\toV
Hu(x)=x-2\langlex,u\rangleu.
This operator reflects the vector
x
u
It is also common to choose a non-unit vector
q\inV
Hq\left(x\right)=x-2
\langlex,q\rangle | |
\langleq,q\rangle |
q.
The Householder operator satisfies the following properties:
V
K
\forall\left(λ,\mu\right)\inK2,\forall\left(x,y\right)\inV2,Hq\left(λx+\muy\right)=λ Hq\left(x\right)+\mu Hq\left(y\right).
K=R
K=C
Over a real or complex vector space, the Householder operator is also known as the Householder transformation.