How can i show that a Householder matrix $U_w$ acts as the identity on the subspace $w$? and that it acts as a reflection on the one-dimensional subspace spanned by w; i.e., $U_w(x) = x$ if $x$ is perpendicular to $w$ and $U_w(w) = -w$.
2026-03-29 15:59:34.1774799974
Householder matrix Uw acts as the identity on the subspace w
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If $\|w\|_2=1$, then with $U=I-2ww^*$, $Uw=(I-2ww^*)w=w-2ww^*w=w-2w=-w$. If $w^*x=0$, $Ux=x-2ww^*x=x-0=x$.