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Unit 1.6.1 Additional homework

Homework 1.6.1.1.

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

  • \(\displaystyle \| e_j \|_2 = \)

  • \(\displaystyle \| e_j \|_1 = \)

  • \(\displaystyle \| e_j \|_\infty = \)

  • \(\displaystyle \| e_j \|_p = \)

Homework 1.6.1.2.

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

  • \(\displaystyle \| I \|_1 = \)

  • \(\displaystyle \| I \|_\infty = \)

  • \(\displaystyle \| I \|_2 = \)

  • \(\displaystyle \| I \|_p = \)

  • \(\displaystyle \| I \|_F = \)

Homework 1.6.1.3.

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

  • \(\displaystyle \| D \|_1 = \)

  • \(\displaystyle \| D \|_\infty = \)

  • \(\displaystyle \| D \|_p = \)

  • \(\displaystyle \| D \|_F = \)

Homework 1.6.1.4.

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: \(\| x_i \|_p \leq \| x \|_p \text{.}\)

Homework 1.6.1.5.

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

  • \(\displaystyle \| A \|_1 = \)

  • \(\displaystyle \| A \|_\infty =\)

  • \(\displaystyle \| A \|_F = \)

Homework 1.6.1.6.

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.

Homework 1.6.1.7.

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 \(\widetilde a_i = ( \widetilde a_i^T )^T \text{.}\)

Homework 1.6.1.8.

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 \(\| x \|_2 = \sqrt{ \vert \chi_0 \vert^2 + \cdots + \vert \chi_{m-1} \vert^2 } \lt \sqrt {\vert \chi_0 \vert + \cdots + \vert \chi_{m-1} \vert } = \sqrt{1} = 1. \)

Homework 1.6.1.9.

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