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Unit 2.5.1 Additional homework

Homework 2.5.1.1.

\(U \in \Cmxm \) 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.

Homework 2.5.1.2.

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 \(U^H B V = A \text{.}\)

Answer

TRUE

Now prove it!

Homework 2.5.1.3.

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 \(A \text{.}\)

Homework 2.5.1.4.

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.

Answer

ALWAYS

Now prove it!

Homework 2.5.1.5.

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.

Homework 2.5.1.6.

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.

Homework 2.5.1.7.

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. \(\kappa_2( A ) = \max_{\| x \|_2 = 1} \| A x \|_2 / \min_{\| x \|_2 = 1} \| A x \|_2\text{.}\)

(Mark all correct answers.)

Homework 2.5.1.8.

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.

Homework 2.5.1.9.

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.

Homework 2.5.1.10.

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 \(A \text{.}\)