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

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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

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

References

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  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