Exercise 6.7.4

Find the singular value decomposition $A=U\mathrm{\Sigma }{V}^{\text{∗}}$. Then we have $W=U{V}^{\text{∗}}$ and $P=V\mathrm{\Sigma }{V}^{\text{∗}}$.

1.

We have

$$W=\left(\begin{array}{cc}\hfill \frac{1}{\sqrt{2}}\hfill & \hfill \frac{1}{\sqrt{2}}\hfill \\ \hfill \frac{1}{\sqrt{2}}\hfill & \hfill -\frac{1}{\sqrt{2}}\hfill \end{array}\right),P=\left(\begin{array}{cc}\hfill \sqrt{2}+\frac{1}{\sqrt{2}}\hfill & \hfill \frac{1}{\sqrt{2}}-\sqrt{2}\hfill \\ \hfill \frac{1}{\sqrt{2}}-\sqrt{2}\hfill & \hfill \sqrt{2}+\frac{1}{\sqrt{2}}\hfill \end{array}\right).$$

2.

We have

$$W=\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right),P=\left(\begin{array}{ccc}\hfill 20\hfill & \hfill 4\hfill & \hfill 0\hfill \\ \hfill 4\hfill & \hfill 20\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right).$$