Chapter 12.3, Problem 5E

Prove that for any constants a and b, the nonzero singular values of the $4\times 2$ matrices are 2a and 2b. Find the reduced SVD in terms of a and b.

$\begin{array}{l}\text{a}\text{. }\left[\begin{array}{c}a\\ -a\\ b\\ -b\end{array}\begin{array}{c}a\\ -a\\ -b\\ b\end{array}\right]\\ \text{b}\text{. }\left[\begin{array}{c}-a\\ a\\ b\\ b\end{array}\begin{array}{c}a\\ -a\\ b\\ b\end{array}\right]\end{array}$