 Solution

If $u=0$, any nonzero $w$, for instance the first Cartesian basis vector $e_1$, gives $H(w)u=0$ and $\alpha =0$. Suppose $u\ne 0$. Choosewhere ${\mathrm{sgn}}_{+}(r)=1$ for $r\ge 0$ and $-1$ otherwise. This choice ensures $w\ne 0$, including when $u$ is already on the first coordinate axis. Since ${\alpha }^2=u^{T}u$,Therefore the Householder reflection satisfiesThe orthogonality of $H(w)$, used here to check length preservation, gives $|\alpha |=\Vert H(w)u{\Vert }_2=\Vert u{\Vert }_2$. The chosen sign also avoids catastrophic cancellation in $w_1=u_1-\alpha$. This is a Householder reflection sending a vector to a coordinate axis with no zero-normal exception.