Householder operator

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In Linear algebra, define the Householder operator as follows.

Let  V\, be a finite dimensional inner product space with unit vector  u\in V Then, the Householder operator is an operator  H_u : V \to V\, defined by

 H_u(x) = x - 2\langle x,u \rangle u\,

where  \langle \cdot, \cdot \rangle is the inner product over V\,. This operator reflects the vector x across a plane given by the normal vector u.[1]

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

The Householder operator has numerous properties such as linearity, being self-adjoint, and is a unitary or orthogonal operator on V.


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