Worksheet 1-2

Worksheet 1-2#

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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, \(\|A\|_F\).

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 \(A\).

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 \(A^TA\).)

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