 Solution

Write $P=\mathbf{a}{\mathbf{a}}^T$. Since $\mathbf{a}$ is a unit vector, $P^2=\mathbf{a}({\mathbf{a}}^T\mathbf{a}){\mathbf{a}}^T=P$ and $P^T=P$. Thus this Householder reflection satisfiesGeometrically, the mirror preserves the component tangent to its plane and reverses the normal component; doing it twice restores the original vector.