Householder operator

In Linear algebra, define the Householder operator as follows.

Let V be a finite dimensional inner product space with unit vector u ∈ V Then, the Householder operator is an operator H u : V → V defined by

where ⟨ ⋅ , ⋅ ⟩ 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.