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 -> V\, defined by

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

where \langle \cdot, \cdot \rangle is the inner product over V\,

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.

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