Householder operator

In Linear algebra, define the Householder operator as follows.

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

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

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

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.