# Worksheet 1-2

Download: [CMSE382-WS1-2.pdf](CMSE382-WS1-2.pdf)

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This is an AI-generated transcript of the worksheet and may contain errors or inaccuracies. Please refer to the original course materials for authoritative content.
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## Worksheet 1-2: Q1

For the matrix $A=\begin{bmatrix} 1 & -2 \\ 3 & 4 \end{bmatrix}$, find

1. The induced $1$-norm, $\|A\|_1$.

2. The induced $\infty$-norm, $\|A\|_\infty$.

3. The Frobenius norm, $\|A\|_F$.

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## Worksheet 1-2: Q2

We are going to determine the induced 2-norm for the matrix $A=\begin{bmatrix} 1 & -2 \\ -2 & 1 \end{bmatrix}$.

1. Compute $A^TA$.

2. Find the eigenvalues of $A^TA$. We generally sort these in decreasing order so that $\lambda_1 \geq \lambda_2$.

3. Find the singular values of $A$.

4. Compute the induced 2-norm of $A$.

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## Worksheet 1-2: Q3

1. Show that for $A \in \mathbb{R}^{m \times n}$, if $\lambda$ is a non-zero eigenvalue of $A^TA$ with eigenvector $v$, then $\lambda$ is a non-zero eigenvalue of $AA^T$ with eigenvector $Av$.

2. Use the previous statement to show that for $A \in \mathbb{R}^{m \times n}$, $\|A\| = \|A^T\|$ for the spectral norm (AKA induced 2-norm).

3. Show that $\|A\|_F^2 = \sum_{i=1}^n \lambda_i(A^TA)$. *(Hint: Consider the trace of $A^TA$.)*