Definitions

# Householder operator

In Linear Algebra, define the Householder operator as follows.

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

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