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

$H\_u(x)\; =\; x\; -\; 2langle\; x,u\; rangle\; u,$

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.