A <Householder matrix> has the form
$$
\boxed{H=I-2\frac{vv^T}{v^Tv},\qquad v\ne0.}
$$
It reflects in the hyperplane perpendicular to $v$. Put $P=vv^T/(v^Tv)$, an <orthogonal projector> satisfying $P^T=P$ and $P^2=P$. Then $H^TH=(I-2P)^2=I-4P+4P^2=I$, proving that every <Householder matrix> is orthogonal.

Solution

Past exam of the numerical linear algebra and approximation course of the University of Oxford / 2008 / 1 / a / ii / Solution

 Solution (source code)