In Linear Algebra, define the Householder operator as follows.

Let be a finite dimensional inner product space

Inner product space

In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...

 with unit vector  Then, the Householder operator is an operator  defined by
where is the inner product over

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

Householder transformation

In linear algebra, a Householder transformation is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. Householder transformations are widely used in numerical linear algebra, to perform QR decompositions and in the first step of the QR algorithm...

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The Householder operator has numerous properties such as linearity, being self-adjoint, and is a unitary or orthogonal operator on V.