Past exam of the mathematics course of the University of Cambridge / 2008 / ib / Paper 2 / 18D / a

Let $P=u{u}^T/(u^{T}u)$, well defined because $u\ne 0$. It is symmetric and satisfies $P^2=P$, so for the Householder reflection $H=I-2P$,Thus $H$ is orthogonal; it reverses the component parallel to $u$ and fixes the perpendicular hyperplane.