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 $u$ .^[1]

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 verifies the following properties:

\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 $K=\mathbb {R}$ , it is orthogonal ; otherwise, if $K = \mathbb{C}$ it is unitary.

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

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