Householder operator

In linear algebra, the Householder operator is defined as follows. Let [math]\displaystyle{ V\, }[/math] be a finite-dimensional inner product space with inner product [math]\displaystyle{ \langle \cdot, \cdot \rangle }[/math] and unit vector [math]\displaystyle{ u\in V }[/math]. Then

[math]\displaystyle{ H_u : V \to V\, }[/math]

is defined by

[math]\displaystyle{ H_u(x) = x - 2\,\langle x,u \rangle\,u\,. }[/math]

This operator reflects the vector [math]\displaystyle{ x }[/math] across a plane given by the normal vector [math]\displaystyle{ u }[/math].

It is also common to choose a non-unit vector [math]\displaystyle{ q \in V }[/math], and normalize it directly in the Householder operator's expression:

[math]\displaystyle{ H_q \left ( x \right ) = x - 2\, \frac{\langle x, q \rangle}{\langle q, q \rangle}\, q \,. }[/math]

Properties

The Householder operator satisfies the following properties:

• It is linear; if [math]\displaystyle{ V }[/math] is a vector space over a field [math]\displaystyle{ K }[/math], then

[math]\displaystyle{ \forall \left ( \lambda, \mu \right ) \in K^2, \, \forall \left ( x, y \right ) \in V^2, \, H_q \left ( \lambda x + \mu y \right ) = \lambda \ H_q \left ( x \right ) + \mu \ H_q \left ( y \right ). }[/math]

• It is self-adjoint.
• If [math]\displaystyle{ K = \mathbb{R} }[/math], then it is orthogonal; otherwise, if [math]\displaystyle{ K = \mathbb{C} }[/math], then it is unitary.

Special cases

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

References

• {{citation | last=Roman | first=Stephen