Exercise 1.8.19

If $S=S^T$ has orthogonal columns $u_1,u_2,u_3$ in $R^3$ of lengths 2,3,4. Then $S=\left[\begin{array}{ccc}\hfill u_1\hfill & \hfill u_2\hfill & \hfill u_3\hfill \end{array}\right]$. Then $U=\left[\begin{array}{ccc}\hfill \frac{u_1}{2}\hfill & \hfill \frac{u_2}{3}\hfill & \hfill \frac{u_3}{4}\hfill \end{array}\right]$ is an orthogonal matrix whose columns are orthognormal. We can easily see that $S=U\left[\begin{array}{ccc}\hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 4\hfill \end{array}\right]I$, so $U$ is the left singular matrix and $I$ is the right singular matrix.