Householder operator

In linear algebra, the Householder operator is defined as follows.^[1] 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 ${\displaystyle u}$.^[2]

It is also common to choose a non-unit vector ${\displaystyle q\in V}$, and normalize it directly in the Householder operator's expression:^[3]

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