= Solution
Rows two and three of $C$ both have zero in column one. Any rotation of these rows therefore preserves those zeros, so $\boxed{d_{21}=d_{31}=0}$. In column two, $d_{32}=c_{22}\sin\theta_3+c_{32}\cos\theta_3$. For $r_3=\sqrt{c_{22}^2+c_{32}^2}>0$, choose
$$
\boxed{\cos\theta_3=\frac{c_{22}}{r_3},\qquad\sin\theta_3=-\frac{c_{32}}{r_3}.}
$$
This gives $d_{32}=0$. Again choose $\theta_3=0$ if both relevant entries are zero. Thus the successive <Givens rotations> have eliminated all entries below the diagonal without recreating earlier ones.
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