Solution
= Solution
For $w\neq0$, the <Householder transformation> is
$$
\boxed{H(w)=I-2\frac{ww^T}{w^Tw}.}
$$
Let $P=ww^T/(w^Tw)$. Then $P^T=P$ and $P^2=P$, since $P$ is the <orthogonal projection matrix> onto the span of $w$. Hence $H^T=H$ and
$$
H^TH=H^2=(I-2P)^2=I-4P+4P^2=I.
$$
Thus $H$ is a <symmetric matrix> and an <orthogonal matrix>. Geometrically it reverses the component parallel to $w$ and fixes the perpendicular <hyperplane>.