Linear Algebra with Applications, 10th Edition

Section 6.4 Exercises

1. For each of the following pairs of vectors z and w, compute (i) $\left|\right|\mathbf{z}\left|\right|$, (ii) $\left|\right|\mathbf{w}\left|\right|$, (iii) $\langle \mathbf{z},\mathbf{w}\rangle$, and (iv) $\langle \mathbf{\text{w, }}\mathbf{z}\rangle$:

   1. $\begin{array}{cc}\mathbf{z}=\left[\begin{array}{c}4+2i\\ 4i\end{array}\right],& \mathbf{w}=\left[\begin{array}{c}-2\\ 2+i\end{array}\right]\end{array}$

   2. $\begin{array}{cc}\mathbf{z}=\left[\begin{array}{c}1+i\\ 2i\\ 3-i\end{array}\right],& \mathbf{w}=\left[\begin{array}{c}2-4i\\ 5\\ 2i\end{array}\right]\end{array}$

2. Let

   $$\begin{array}{cc}{\mathbf{z}}_1=\left[\begin{array}{c}\frac{{\displaystyle \begin{array}{c}1+i\end{array}}}{{\displaystyle 2}}\\ \frac{{\displaystyle 1-i}}{{\displaystyle 2}}\end{array}\right]& \begin{array}{cc}\text{and}& {\mathbf{z}}_2\left[\begin{array}{c}\frac{{\displaystyle i}}{{\displaystyle \sqrt{2}}}\\ -\frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}\end{array}\right]\end{array}\end{array}$$

   1. Show that $\{{\mathbf{z}}_1,{\mathbf{z}}_2\}$ is an orthonormal set in ${ℂ}^2$.

   2. Write the vector $\mathbf{z}=\left[\begin{array}{c}2+4i\\ -2i\end{array}\right]$ as a linear combination of ${\mathbf{z}}_1$ and ${\mathbf{z}}_2$.

3. Let $\{{\text{u}}_1,{\text{u}}_2\}$ be an orthonormal basis for ${ℂ}^2$, and let $\mathbf{z}=(4+2i){\mathbf{\text{u}}}_1+(6-5i){\mathbf{\text{u}}}_2$.

   1. What are the values of ${\mathbf{u}}_1^H\mathbf{z},{\mathbf{z}}^H{\mathbf{u}}_1,{\mathbf{u}}_2^H\mathbf{z}$, and ${\mathbf{z}}^H{\mathbf{u}}_2?$

   2. Determine the value of $\Vert \mathbf{z}\Vert$.

4. Which of the matrices that follow are Hermitian? Normal?

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

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

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

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

   5. $\left[\begin{array}{ccc}0& i& 1\\ i& 0& -2+i\\ -1& 2+i& 0\end{array}\right]$

   6. $\left[\begin{array}{ccc}3& 1+i& i\\ 1-i& 1& 3\\ -i& 3& 1\end{array}\right]$

5. Find an orthogonal or unitary diagonalizing matrix for each of the following:

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

   2. $\left[\begin{array}{cc}1& 3+i\\ 3-i& 4\end{array}\right]$

   3. $\left[\begin{array}{ccc}\hfill 2& i& 0\\ \hfill -i& 2& 0\\ \hfill 0& 0& 2\end{array}\right]$

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

   5. $[$