Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / P / P4 / b / Solution

If $x\in S_v$, then $v^{T}x=0$ and $Px=0$, so $\boxed{H_{v}x=x}$. For arbitrary $x$, $v^{T}P=v^T$, givingThus the Householder map fixes the hyperplane orthogonal to $v$ and reverses the component in its normal direction: it is a reflection, not merely an arbitrary orthogonal map.