Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / P / P4 / a

For nonzero $w$, the Householder transformation isThe rank-one matrix $P=w{w}^T/(w^{T}w)$ is symmetric and satisfies $P^2=P$. Consequently $H^T=H$ and $H^{T}H=(I-2P{)}^2=I-4P+4P^2=I$. Thus the matrix is symmetric and orthogonal; geometrically it reflects the component along $w$ and fixes its perpendicular hyperplane.