Householder operator

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

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

is defined by

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

This operator reflects the vector $x$ across a plane given by the normal vector $u$.

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

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

Properties

The Householder operator satisfies the following properties:

• It is linear; if $V$ is a vector space over a field $K$, then

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

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

Special cases

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

References

• Roman, Stephen (2008), Advanced Linear Algebra, Graduate Texts in Mathematics (Third ed.), Springer, ISBN 978-0-387-72828-5