In Linear algebra, define the Householder operator as follows.
Let be a finite dimensional inner product space with unit vector Then, the Householder operator is an operator defined by
where is the inner product over . This operator reflects the vector across a plane given by the normal vector .
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.