Householder operator

In linear algebra, the Householder operator  is defined as follows. Let $$ V\, $$ be a finite-dimensional inner product space with inner product $$ \langle \cdot, \cdot \rangle $$ and unit vector $$ u\in V$$. Then
 * $$ H_u : V \to V\,$$

is defined by
 * $$ H_u(x) = x - 2\,\langle x,u \rangle\,u\,.$$

This operator reflects the vector $$x$$ across a plane given by the normal vector $$u$$.

It is also common to choose a non-unit vector $$q \in V$$, and normalize it directly in the Householder operator's expression:
 * $$H_q \left ( x \right ) = x - 2\, \frac{\langle x, q \rangle}{\langle q, q \rangle}\, q \,.$$

Properties
The Householder operator satisfies the following properties:


 * It is linear; if $$V$$ is a vector space over a field $$K$$, then
 * $$\forall \left ( \lambda, \mu \right ) \in K^2, \, \forall \left ( x, y \right ) \in V^2, \, H_q \left ( \lambda x + \mu y \right ) = \lambda \ H_q \left ( x \right ) + \mu \ H_q \left ( y \right ).$$


 * It is self-adjoint.
 * If $$K = \mathbb{R}$$, then it is orthogonal; otherwise, if $$K = \mathbb{C}$$, then it is unitary.

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