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

A real square orthogonal matrix $Q$ satisfies $\boxed{Q^{T}Q=Q{Q}^T=I}$. Equivalently its columns form an orthonormal basis. In particular it preserves the Euclidean norm, since $\Vert Qx{\Vert }_2^2=x^T{Q}^{T}Qx=\Vert x{\Vert }_2^2$.

A Householder matrix has the formIt reflects in the hyperplane perpendicular to $v$. Put $P=v{v}^T/(v^{T}v)$, an orthogonal projector satisfying $P^T=P$ and $P^2=P$. Then $H^{T}H=(I-2P{)}^2=I-4P+4P^2=I$, proving that every Householder matrix is orthogonal.