Householder operator

In Linear algebra, define the Householder operator as follows.

Let $V\,$ be a finite dimensional inner product space with unit vector $u\in V$ Then, the Householder operator is an operator $H_u : V \to V\,$ defined by

H_u(x) = x - 2\langle x,u \rangle u\,

where $\langle \cdot, \cdot \rangle$ is the inner product over $V\,$ . This operator reflects the vector $x$ across a plane given by the normal vector $u$ .^[1]

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.

References

↑ Methods of Applied Mathematics for Engineers and Scientist. Cambridge University Press. pp. Section E.4.11. ISBN 9781107244467.