Householder operator

In linear algebra, the Householder operator is defined as follows. 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 .[1]

It is also common to choose a non-unit vector , and normalize it directly in the Householder operator's expression

Properties

The Householder operator verifies the following properties:

  • it is linear ; if is a vector space over a field , then

Special cases

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

gollark: Would be nice, but isn't there yet.
gollark: And "negotiates resources" how?
gollark: What? That makes even less sense. So some unlucky people are in "production", and everyone else does ???.
gollark: And manage services.
gollark: And take in taxes.

References

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


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