Linear Algebra with Applications, 10th Edition

Section 6.5 Exercises

1. Show that A and $A^T$ have the same nonzero singular values. How are their singular value decompositions related?

2. Use the method of Example 1 to find the singular value decomposition of each of the following matrices:

   1. $\left[\begin{array}{cc}1& 1\\ 2& 2\end{array}\right]$

   2. $\left[\begin{array}{cc}2& -2\\ 1& 2\end{array}\right]$

   3. $\left[\begin{array}{cc}1& 3\\ 3& 1\\ 0& 0\\ 0& 0\end{array}\right]$

   4. $\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 1\\ 0& 1& 2\\ 0& 0& 0\end{array}\right]$

3. For each of the matrices in Exercise 2:

   1. determine the rank.

   2. find the closest (with respect to the Frobenius norm) matrix of rank 1.

4. Let

   $$A=\left[\begin{array}{ccc}\hfill -2& \hfill 8& \hfill 20\\ \hfill 14& \hfill 19& \hfill 10\\ \hfill 2& \hfill -2& \hfill 1\end{array}\right]=\left[\begin{array}{ccc}\frac{3}{5}& \hfill -\frac{4}{5}& 0\\ \frac{4}{5}& \hfill \frac{3}{5}& 0\\ 0& \hfill 0& 1\end{array}\right]\left[\begin{array}{ccc}\hfill 30& \hfill 0& 0\\ \hfill 0& \hfill 15& 0\\ \hfill 0& \hfill 0& 3\end{array}\right]\left[\begin{array}{ccc}\frac{1}{3}& \hfill \frac{2}{3}& \hfill \frac{2}{3}\\ \frac{2}{3}& \hfill \frac{1}{3}& \hfill -\frac{2}{3}\\ \frac{2}{3}& \hfill -\frac{2}{3}& \hfill \frac{1}{3}\end{array}\right]$$

   Find the closest (with respect to the Frobenius norm) matrices of rank 1 and rank 2 to A.

5. The matrix

   $$A=\left[\begin{array}{ccc}2& 5& 4\\ 6& 3& 0\\ 6& 3& 0\\ 2& 5& 4\end{array}\right]$$

   has singular value decomposition

   $$\left[\begin{array}{cccc}\frac{1}{2}& \hfill \frac{1}{2}& \hfill \frac{1}{2}& \hfill \frac{1}{2}\\ \frac{1}{2}& \hfill -\frac{1}{2}& \hfill -\frac{1}{2}& \hfill \frac{1}{2}\\ \frac{1}{2}& \hfill -\frac{1}{2}& \hfill \frac{1}{2}& \hfill -\frac{1}{2}\\ \frac{1}{2}& \hfill \frac{1}{2}& \hfill -\frac{1}{2}& \hfill -\frac{1}{2}\end{array}\right]\left[\begin{array}{ccc}12& 0& 0\\ 0& 6& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{ccc}\hfill \frac{2}{3}& \hfill \frac{2}{3}& \frac{1}{3}\\ \hfill -\frac{2}{3}& \hfill \frac{1}{3}& \frac{2}{3}\\ \hfill \frac{1}{3}& \hfill -\frac{2}{3}& \frac{2}{3}\end{array}\right]$$

   1. Use the singular ...