###### Homework1.6.1.1.

For $e_j \in \Rn$ (a standard basis vector), compute

• $\| e_j \|_2 =$

• $\| e_j \|_1 =$

• $\| e_j \|_\infty =$

• $\| e_j \|_p =$

###### Homework1.6.1.2.

For $I \in \Rnxn$ (the identity matrix), compute

• $\| I \|_1 =$

• $\| I \|_\infty =$

• $\| I \|_2 =$

• $\| I \|_p =$

• $\| I \|_F =$

###### Homework1.6.1.3.

Let $D = \left( \begin{array}{c c c c} \delta_0 \amp 0 \amp \cdots \amp 0 \\ 0 \amp \delta_1 \amp \cdots \amp 0 \\ \vdots \amp \ddots \amp \ddots \amp 0 \\ 0 \amp 0 \amp \cdots \amp \delta_{n-1} \end{array} \right)$ (a diagonal matrix). Compute

• $\| D \|_1 =$

• $\| D \|_\infty =$

• $\| D \|_p =$

• $\| D \|_F =$

###### Homework1.6.1.4.

Let $x = \left( \begin{array}{c} x_0 \\ \hline x_1 \\ \hline \vdots \\ \hline x_{N-1} \end{array} \right)$ and $1 \leq p \lt \infty$ or $p = \infty \text{.}$

ALWAYS/SOMETIMES/NEVER: $\| x_i \|_p \leq \| x \|_p \text{.}$

###### Homework1.6.1.5.

For

\begin{equation*} A = \left( \begin{array}{r r r} 1 \amp 2 \amp -1 \\ -1 \amp 1 \amp 0 \end{array} \right). \end{equation*}

compute

• $\| A \|_1 =$

• $\| A \|_\infty =$

• $\| A \|_F =$

###### Homework1.6.1.6.

For $A \in \C^{m \times n}$ define

\begin{equation*} \| A \| = \sum_{i=0}^{m-1}\sum_{j=0}^{n-1} \vert \alpha_{i,j} \vert = \sum \left( \begin{array}{c c c} \vert \alpha_{0,0} \vert , \amp \cdots , \amp \vert \alpha_{0,n-1} \vert, \\ \vdots \amp \amp \vdots \\ \vert \alpha_{m-1,0} \vert , \amp \cdots , \amp \vert \alpha_{m-1,n-1} \vert \end{array} \right) . \end{equation*}
• TRUE/FALSE: This function is a matrix norm.

• How can you relate this norm to the vector 1-norm?

• TRUE/FALSE: For this norm, $\| A \| = \| A^H \| \text{.}$

• TRUE/FALSE: This norm is submultiplicative.

###### Homework1.6.1.7.

Let $A \in \mathbb C^{m \times n} \text{.}$ Partition

\begin{equation*} A = \left( \begin{array}{c | c | c | c} a_0 \amp a_1 \amp \cdots \amp a_{n-1} \end{array} \right) = \left( \begin{array}{c} \widetilde a_0^T \\ \widetilde a_1^T \\ \vdots \\ \widetilde a_{m-1}^T \end{array} \right). \end{equation*}

Prove that

• $\| A \|_F = \| A^T \|_F \text{.}$

• $\| A \|_F = \sqrt{ \| a_0 \|_2^2 + \| a_1 \|_2^2 + \cdots + \| a_{n-1} \|_2^2 } \text{.}$

• $\| A \|_F = \sqrt{ \| \widetilde a_0 \|_2^2 + \| \widetilde a_1 \|_2^2 + \cdots + \| \widetilde a_{m-1} \|_2^2 } \text{.}$

Note that here $\widetilde a_i = ( \widetilde a_i^T )^T \text{.}$

###### Homework1.6.1.8.

Let $x \in \Rm$ with $\| x \|_1 = 1 \text{.}$

TRUE/FALSE: $\| x \|_2 = 1$ if and only if $x = \pm e_j$ for some $j \text{.}$

Solution

Obviously, if $x = e_j$ then $\| x \|_1 = \| x |_2 = 1 \text{.}$

Assume $x \neq e_j \text{.}$ Then $\vert \chi_i \vert \lt 1$ for all $i \text{.}$ But then $\| x \|_2 = \sqrt{ \vert \chi_0 \vert^2 + \cdots + \vert \chi_{m-1} \vert^2 } \lt \sqrt {\vert \chi_0 \vert + \cdots + \vert \chi_{m-1} \vert } = \sqrt{1} = 1.$

###### Homework1.6.1.9.

Prove that if $\| x \|_\nu \leq \beta \| x \|_\mu$ is true for all $x \text{,}$ then $\| A \|_\nu \leq \beta \| A \|_{\mu,\nu} \text{.}$