Householder operator

In linear algebra, the Householder operator is defined as follows. Let formula_1 be a finite dimensional inner product space with inner product formula_2 and unit vector formula_3. Then
is defined by
This operator reflects the vector formula_6 across a plane given by the normal vector formula_7.
It is also common to choose a non-unit vector formula_8, and normalize it directly in the Householder operator's expression
Properties.
The Householder operator verifies the following properties:
Special cases.
Over a real or complex vector space, the Householder operator is also known as the Householder transformation.