Householder operator

In linear algebra, the Householder operator is defined as follows. Let ${\displaystyle V\,}$ be a finite dimensional inner product space with inner product ${\displaystyle \langle \cdot ,\cdot \rangle }$ and unit vector ${\displaystyle u\in V}$. Then

${\displaystyle H_{u}:V\to V\,}$

is defined by

${\displaystyle H_{u}(x)=x-2\,\langle x,u\rangle \,u\,}$

This operator reflects the vector ${\displaystyle x}$ across a plane given by the normal vector ${\displaystyle u}$.[1]

It is also common to choose a non-unit vector ${\displaystyle q\in V}$, and normalize it directly in the Householder operator's expression

${\displaystyle H_{q}\left(x\right)=x-2\,{\frac {\langle x,q\rangle }{\langle q,q\rangle }}\,q\,}$

Properties

The Householder operator verifies the following properties:

• it is linear ; if ${\displaystyle V}$ is a vector space over a field ${\displaystyle K}$, then

${\displaystyle \forall \left(\lambda ,\mu \right)\in K^{2},\,\forall \left(x,y\right)\in V^{2},\,H_{q}\left(\lambda x+\mu y\right)=\lambda \ H_{q}\left(x\right)+\mu \ H_{q}\left(y\right)}$

• self-adjoint
• if ${\displaystyle K=\mathbb {R} }$, it is orthogonal ; otherwise, if ${\displaystyle K=\mathbb {C} }$ it is unitary.

Special cases

Over a real or complex vector space, the Householder operator is also known as the Householder transformation.

References

1. Methods of Applied Mathematics for Engineers and Scientist. Cambridge University Press. pp. Section E.4.11. ISBN 9781107244467.