Linear Algebra with Applications, 10th Edition

Section 7.4 Exercises

1. Determine $\Vert {\cdot \Vert }_F$, $\Vert {\cdot \Vert }_{\infty }$, and $\Vert {\cdot \Vert }_1$ for each of the following matrices:

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

   2. $\left[\begin{array}{cc}\hfill 1& 4\\ \hfill -2& 2\end{array}\right]$

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

   4. $\left[\begin{array}{ccc}0& 5& 1\\ 2& 3& 1\\ 1& 2& 2\end{array}\right]$

   5. $\left[\begin{array}{ccc}5& 0& 5\\ 4& 1& 0\\ 3& 2& 1\end{array}\right]$

2. Let

   $$\begin{array}{ccc}A=\left[\begin{array}{cc}2& \hfill 0\\ 0& \hfill -2\end{array}\right]& \text{and}& \mathbf{\text{x}}=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\end{array}$$

   and set

   $$f(x_1,x_2)={\Vert A\mathbf{\text{x}}\Vert }_2/\Vert \mathbf{\text{x}}{\Vert }_2$$

   Determine the value of $\Vert {A\Vert }_2$ by finding the maximum value of f for all $(x_1,x_2)\ne (0,0)$.

3. Let

   $$A=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$$

   Use the method of Exercise 2 to determine the value of ${\Vert A\Vert }_2$

4. Let

   $$D=\left[\begin{array}{cccc}3& \hfill 0& \hfill 0& 0\\ 0& \hfill -5& \hfill 0& 0\\ 0& \hfill 0& \hfill -2& 0\\ 0& \hfill 0& \hfill 0& 4\end{array}\right]$$

   1. Compute the singular value decomposition of D.

   2. Find the value of ${\Vert D\Vert }_2$.

5. Show that if D is an $n\times n$ diagonal matrix, then

   $${\Vert D\Vert }_2=\underset{1\le i\le n}{\text{max}}\left(\left|d_{ii}\right|\right)$$

6. If D is an $n\times n$ diagonal matrix, how do the values of ${\Vert D\Vert }_1$, ${\Vert D\Vert }_2$, and ${\Vert D\Vert }_{\infty }$ compare? Explain your answers.

7. Let I denote the $n\times n$ identity matrix. Determine the values of ${\Vert I\Vert }_1$, ${\Vert I\Vert }_{\infty }$, and ${\Vert I\Vert }_F$.

8. Let ${\Vert \cdot \Vert }_M$ denote a matrix norm ...