Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 3 / 19D / b / Solution

For a nonzero real vector $v$, its Householder reflection isLet $P=v{v}^T/(v^{T}v)$. Then $P^T=P$ and $P^2=P$, soThus $H$ is an orthogonal matrix, and it is also symmetric with $H^{-1}=H$. It sends $v$ to $-v$ and fixes every vector perpendicular to $v$, explaining the reflection in the hyperplane $v^{\perp }$. For a target reduction of a vector $a$ to $\Vert a\Vert e_1$, choose $v=a-\Vert a\Vert e_1$ when this vector is nonzero; an alternative sign avoids cancellation in numerical implementations.