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 $H\_u\; :\; V\; o\; V,$ defined by:$H\_u(x)\; =\; x\; -\; 2langle\; x,u\; angle\; u,$ where $langle\; cdot,\; cdot\; angle$ 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.