In linear algebra, the Householder operator is defined as follows.[1] Let be a finite-dimensional inner product space with inner product and unit vector . Then
is defined by
This operator reflects the vector across a plane given by the normal vector .[2]
It is also common to choose a non-unit vector , and normalize it directly in the Householder operator's expression:[3]
Properties
editThe Householder operator satisfies the following properties:
- It is self-adjoint.
- If , then it is orthogonal; otherwise, if , then it is unitary.
Special cases
editOver a real or complex vector space, the Householder operator is also known as the Householder transformation.
References
edit- ^ Roman 2008, p. 243-244
- ^ Methods of Applied Mathematics for Engineers and Scientist. Cambridge University Press. pp. Section E.4.11. ISBN 9781107244467.
- ^ Roman 2008, p. 244
- Roman, Stephen (2008), Advanced Linear Algebra, Graduate Texts in Mathematics (Third ed.), Springer, ISBN 978-0-387-72828-5