###### Homework2.5.1.1.

$U \in \Cmxm$ is unitary if and only if $( U x )^H ( U y ) = x^H y$ for all $x, y \in \Cm \text{.}$

Hint

Revisit the proof of Homework 2.2.4.6.

###### Homework2.5.1.2.

Let $A, B \in \C^{m \times n} \text{.}$ Furthermore, let $U \in \C^{m \times m}$ and $V \in \C^{n \times n}$ be unitary.

TRUE/FALSE: $U A V^H = B$ iff $U^H B V = A \text{.}$

TRUE

Now prove it!

###### Homework2.5.1.3.

Prove that nonsingular $A \in \Cnxn$ has condition number $\kappa_2(A)=1$ if and only if $A = \sigma Q$ where $Q$ is unitary and $\sigma \in \R$ is positive.

Hint

Use the SVD of $A \text{.}$

###### Homework2.5.1.4.

Let $U \in \Cmxm$ and $V \in \Cnxn$ be unitary.

ALWAYS/SOMETIMES/NEVER: The matrix $\left( \begin{array}{c | c} U \amp 0 \\ \hline 0 \amp V \end{array} \right)$ is unitary.

ALWAYS

Now prove it!

###### Homework2.5.1.5.

Matrix $A \in \mathbb R^{m \times m}$ is a stochastic matrix if and only if it is nonnegative (all its entries are nonnegative) and the entries in its columns sum to one: $\sum_{0 \leq i \lt m} \alpha_{i,j} = 1 \text{.}$ Such matrices are at the core of Markov processes. Show that a matrix $A$ is both unitary matrix and a stochastic matrix if and only if it is a permutation matrix.

###### Homework2.5.1.6.

Show that if $\| \cdots \|$ is a norm and $A$ is nonsingular, then $\| \cdots \|_{A^{-1}}$ defined by $\| x \|_{A^{-1}} = \| A^{-1} x \|$ is a norm.

Interpret this result in terms of the change of basis of a vector.

###### Homework2.5.1.7.

Let $A \in \Cmxm$ be nonsingular and $A = U \Sigma V^H$ be its SVD with

\begin{equation*} \Sigma = \left( \begin{array}{ c | c | c | c} \sigma_0 \amp 0 \amp \cdots \amp 0 \\ \hline 0 \amp \sigma_1 \amp \cdots \amp 0 \\ \hline \vdots \amp \vdots \amp \ddots \amp \vdots \\ \hline 0 \amp 0 \amp \cdots \amp \sigma_{m-1} \end{array} \right) \end{equation*}

The condition number of $A$ is given by (mark all correct answers):

1. $\kappa_2( A ) = \| A \|_2 \| A^{-1} \|_2 \text{.}$
2. $\kappa_2( A ) = \sigma_0 / \sigma_{m-1} \text{.}$
3. $\kappa_2( A ) = u_0^H A v_0 / u_{m-1} ^H A v_{m-1} \text{.}$
4. $\kappa_2( A ) = \max_{\| x \|_2 = 1} \| A x \|_2 / \min_{\| x \|_2 = 1} \| A x \|_2\text{.}$

###### Homework2.5.1.8.

Theorem 2.2.4.4 stated:

If $A \in \Cmxm$ preserves length ($\| A x \|_2 = \| x \|_2$ for all $x \in \Cm$), then $A$ is unitary.

Give an alternative proof using the SVD.

###### Homework2.5.1.9.

In Homework 1.3.7.2 you were asked to prove that $\| A \|_2 \leq \| A \|_F$ given $A \in \Cmxn \text{.}$ Give an alternative proof that leverages the SVD.

###### Homework2.5.1.10.

In Homework 1.3.7.3, we skipped how the 2-norm bounds the Frobenius norm. We now have the tools to do so elegantly: Prove that, given $A \in \Cmxn \text{,}$

\begin{equation*} \| A \|_F \leq \sqrt{r} \| A \|_2, \end{equation*}

where $r$ is the rank of matrix $A \text{.}$