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Householder operator

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Concept in mathematics

In linear algebra, the Householder operator is defined as follows. Let V {\displaystyle V\,} be a finite-dimensional inner product space with inner product , {\displaystyle \langle \cdot ,\cdot \rangle } and unit vector u V {\displaystyle u\in V} . Then

H u : V V {\displaystyle H_{u}:V\to V\,}

is defined by

H u ( x ) = x 2 x , u u . {\displaystyle H_{u}(x)=x-2\,\langle x,u\rangle \,u\,.}

This operator reflects the vector x {\displaystyle x} across a plane given by the normal vector u {\displaystyle u} .

It is also common to choose a non-unit vector q V {\displaystyle q\in V} , and normalize it directly in the Householder operator's expression:

H q ( x ) = x 2 x , q q , q q . {\displaystyle H_{q}\left(x\right)=x-2\,{\frac {\langle x,q\rangle }{\langle q,q\rangle }}\,q\,.}

Properties

The Householder operator satisfies the following properties:

  • It is linear; if V {\displaystyle V} is a vector space over a field K {\displaystyle K} , then
( λ , μ ) K 2 , ( x , y ) V 2 , H q ( λ x + μ y ) = λ   H q ( x ) + μ   H q ( y ) . {\displaystyle \forall \left(\lambda ,\mu \right)\in K^{2},\,\forall \left(x,y\right)\in V^{2},\,H_{q}\left(\lambda x+\mu y\right)=\lambda \ H_{q}\left(x\right)+\mu \ H_{q}\left(y\right).}
  • It is self-adjoint.
  • If K = R {\displaystyle K=\mathbb {R} } , then it is orthogonal; otherwise, if K = C {\displaystyle K=\mathbb {C} } , then it is unitary.

Special cases

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

References

  1. Roman 2008, p. 243-244
  2. Methods of Applied Mathematics for Engineers and Scientist. Cambridge University Press. 28 June 2013. pp. Section E.4.11. ISBN 9781107244467.
  3. Roman 2008, p. 244


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