Jump to content

Householder operator

From Wikipedia, the free encyclopedia

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

[edit]

The Householder operator satisfies the following properties:

  • It is linear; if is a vector space over a field , then
  • It is self-adjoint.
  • If , then it is orthogonal; otherwise, if , then it is unitary.

Special cases

[edit]

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

References

[edit]
  1. ^ Roman 2008, p. 243-244
  2. ^ Methods of Applied Mathematics for Engineers and Scientist. Cambridge University Press. pp. Section E.4.11. ISBN 9781107244467.
  3. ^ Roman 2008, p. 244