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

Let

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 ${\mathbf{z}}_2$.

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 $\Vert \mathbf{z}\Vert$.

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

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

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

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

Show that the diagonal entries of a Hermitian matrix must be real.

Let A be an $n\times n$ Hermitian matrix and let x be a vector in ${ℂ}^n$. Show that if $c={\mathbf{x}}^{H}A\mathbf{x}$, then c is real.

Let A be an Hermitian matrix and let $B=iA$. Show that B is skew Hermitian.

Let A and C be matrices in ${ℂ}^{m\times n}$ and let $B\in {ℂ}^{n\times r}$. Prove each of the following rules:

1. $(A^H{)}^H=A$

2. $(\alpha A+\beta C{)}^H=\overline{\alpha }{A}^H+\overline{\beta }{C}^H$

3. $(AB{)}^H=B^H{A}^H$

Let A and B be Hermitian matrices. Answer true or false for each of the statements that follow. In each case, explain or prove your answer.

1. The eigenvalues of AB are all real.

2. The eigenvalues of ABA are all real.

Show that

defines an inner product on ${ℂ}^n$.

Let x, y, and z be vectors in ${ℂ}^n$ and let $\alpha$ and $\beta$ be complex scalars. Show that

Let $\{{\mathbf{u}}_1,\mathrm{\dots },{\mathbf{u}}_n\}$ be an orthonormal basis for a complex inner product space V, and let

Show that

Given that

find a matrix B such that $B^{H}B=A$.

Let U be a unitary matrix. Prove that

1. U is normal.

2. $\Vert U_{\mathbf{x}}\Vert =\Vert \mathbf{x}\Vert$ for all $\mathbf{x}\in {ℂ}^n$.

3. if $\lambda$ is an eigenvalue of U, then $\left|\lambda \right|=1$

Let u be a unit vector in ${ℂ}^n$ and define $U=I-2\mathbf{u}{\mathbf{u}}^H$. Show that U is both unitary and Hermitian and, consequently, is its own inverse.

Show that if a matrix U is both unitary and Hermitian, then any eigenvalue of U must equal either 1 or −1.

Let A be a $2\times 2$ matrix with Schur decomposition $UT{U}^H$ and suppose that $t_{12}\ne 0$. Show that

1. the eigenvalues of A are ${\lambda }_1=t_{11}$ and ${\lambda }_2=t_{22}$.

2. ${\mathbf{u}}_1$ is an eigenvector of A belonging to ${\lambda }_1=t_{11}$.

3. ${\mathbf{u}}_2$ is not an eigenvector of A belonging to ${\lambda }_2=t_{22}$.

Let A be a $5\times 5$ matrix with real entries. Let $A=QT{Q}^T$ be the real Schur decomposition of A, where T is a block matrix of the form given in equation $\left(2\right)$. What are the possible block structures for T in each of the following cases?

1. All of the eigenvalues of A are real.

2. A has three real eigenvalues and two complex eigenvalues.

3. A has one real eigenvalue and four complex eigenvalues.

Let A be a $n\times n$ matrix with Schur decomposition $UT{U}^H$. Show that if the diagonal entries of T are all distinct, then there is an upper triangular matrix R such that $X=UR$ diagonalizes A.

Show that $M=A+iB$ (where A and B are real matrices) is skew Hermitian if and only if A is skew symmetric and B is symmetric.

Show that if A is skew Hermitian and $\lambda$ is an eigenvalue of A, then $\lambda$ is purely imaginary (i.e., $\lambda =bi$, where b is real).

Show that if A is a normal matrix, then each of the following matrices must also be normal:

1. $A^H$

2. $I+A$

3. $A^2$

Let A be a real $2\times 2$ matrix with the property that $a_{21}{a}_{12}>0$, and let

Compute $B=SA{S}^{-1}$. What can you conclude about the eigenvalues and eigenvectors of B? What can you conclude about the eigenvalues and eigenvectors of A? Explain.

Let $p\left(x\right)=-x^3+c{x}^2+(c+3)x+1$, where c is a real number. Let

and let

1. Compute $A^{-1}CA$

2. Show that C is the companion matrix of $p\left(x\right)$ and use the result from part $\left(\text{a}\right)$ to prove that $p\left(x\right)$ will have only real roots regardless of the value of c.

Let A be a Hermitian matrix with eigenvalues ${\lambda }_1,\mathrm{\dots },{\lambda }_n$ and orthonormal eigenvectors ${\mathbf{u}}_1,\mathrm{\dots },{\mathbf{u}}_n$. Show that

Let

Write A as a sum ${\lambda }_1{\mathbf{u}}_1{\mathbf{u}}_1^T+{\lambda }_2{\mathbf{u}}_2{\mathbf{u}}_2^T$, where ${\lambda }_1$ and ${\lambda }_2$ are eigenvalues and ${\mathbf{u}}_1$ and ${\mathbf{u}}_2$ are orthonormal eigenvectors.

Let A be a Hermitian matrix with eigenvalues ${\lambda }_1\ge {\lambda }_2\ge \mathrm{\dots }\ge {\lambda }_n$ and orthonormal eigenvectors ${\mathbf{u}}_1,\mathrm{\dots },{\mathbf{u}}_n$. For any nonzero vector x in ${ℝ}^n$, the Rayleigh quotient $\rho \left(\mathbf{x}\right)$ is defined by

1. If $\mathbf{x}=c_1{\mathbf{u}}_1+\mathrm{\dots }+c_n{\mathbf{u}}_n$, show that

   $$\rho \left(\mathbf{\text{x}}\right)=\frac{{\displaystyle |c_1{|}^2{\lambda }_1+|c_2{|}^2{\lambda }_2+\mathrm{\dots }+|c_n{|}^2{\lambda }_n}}{{\displaystyle \Vert \mathbf{c}{\Vert }^2}}$$

2. Show that

   $${\lambda }_n\le \rho \left(\mathbf{\text{x}}\right)\le {\lambda }_1$$

3. Show that

   $$\begin{array}{ccc}\underset{\mathbf{x}\ne 0}{\text{max}}\rho \left(\mathbf{x}\right)={\lambda }_1& \text{and}& \underset{\mathbf{x}\ne 0}{\text{min}}\rho \left(\mathbf{x}\right)={\lambda }_n\end{array}$$

Given $A\in {ℝ}^{m\times m},B\in {ℝ}^{n\times n},C\in {ℝ}^{m\times n}$, the equation

is known as Sylvester’s equation. An m × n matrix X is said to be a solution if it satisfies $\left(3\right)$.

1. Show that if B has Schur decomposition $B=UT{U}^H$, then Sylvester’s equation can be transformed into an equation of the form $AY-YT=G$, where $Y=XU$ and $G=CU$.

2. Show that

   $$\begin{array}{cc}\hfill (A-t_{11}I){\mathbf{\text{y}}}_1& ={\mathbf{\text{g}}}_1\hfill \\ \hfill (A-t_{jj}I){\mathbf{\text{y}}}_j& ={\mathbf{\text{g}}}_j+\mathrm{\sum }_{i=1}^{j-1}{t}_{ij}{\mathbf{\text{y}}}_j,j=2,\mathrm{\dots },n\hfill \end{array}$$

3. Show that if A and B have no common eigenvalues, then Sylvester’s equation has a solution.