# Householder operator

In Linear algebra, define the Householder operator as follows.

Let ${\displaystyle V\,}$ be a finite dimensional inner product space with unit vector ${\displaystyle u\in V}$ Then, the Householder operator is an operator ${\displaystyle H_{u}:V\to V\,}$ defined by

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

where ${\displaystyle \langle \cdot ,\cdot \rangle }$ is the inner product over ${\displaystyle V\,}$. This operator reflects the vector ${\displaystyle x}$ across a plane given by the normal vector ${\displaystyle u}$.[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

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