= Solution
The second <matrix> acts by
$$
Q=\begin{pmatrix}0&1&0\\1&0&0\\0&0&1\end{pmatrix},\qquad Q\mathbf x=(x_2,x_1,x_3)^T.
$$
Applying $P$ from part (e) then gives
$$
\boxed{\mathbf y=PQ\mathbf x=(-x_1,-x_2,x_3)^T,\qquad PQ=\operatorname{diag}(-1,-1,1).}
$$
The <Householder transformation> $Q$ reflects in the <plane> $x_1-x_2=0$. Its <normal vector> $\mathbf q$ is perpendicular to $\mathbf p$, so the two reflecting <planes> meet at a right angle along the $x_3$ axis. Their composition is \b[a rotation through $\pi$ about the $x_3$ axis], as is also immediate from the displayed <rotation matrix>.
Back to article page
