Past exam of the options II course of the University of Oxford / 2011 / P / P4 / a

For $w\ne 0$, the Householder transformation isLet $P=w{w}^T/(w^{T}w)$. Then $P^T=P$ and $P^2=P$, since $P$ is the orthogonal projection matrix onto the span of $w$. Hence $H^T=H$ andThus $H$ is a symmetric matrix and an orthogonal matrix. Geometrically it reverses the component parallel to $w$ and fixes the perpendicular hyperplane.