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[]

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