Biostat 216 Homework 2

Due Oct 11 @ 11:59pm

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1 Q1. Sub-multiplicity of Frobenius norm

Show the matrix norm property \[ \|\mathbf{A} \mathbf{B}\|_{\text{F}} \le \|\mathbf{A}\|_{\text{F}} \|\mathbf{B}\|_{\text{F}} \] for the Frobenius norm. Hint: Cauchy-Schwartz inequality.

BV exercise 6.14 is a special case of this result.

2 Q2. Induced matrix norm

For any vector norm \(\|\mathbf{x}\|\) on \(\mathbb{R}^m\) and \(\mathbb{R}^n\), there is an induced matrix norm \(\|\mathbf{A}\|\) on \(m \times n\) matrices defined by \[ \|\mathbf{A}\| = \sup_{\mathbf{x} \ne \mathbf{0}} \frac{\|\mathbf{A} \mathbf{x}\|}{\|\mathbf{x}\|} = \sup_{\|\mathbf{x}\|=1} \|\mathbf{A} \mathbf{x}\|. \]

1. Show the second equality in the above equation.

2. Show the four properties (positive definiteness, homogeneity, triangle inequality, sub-multiplicity) for the induced matrix norm.

3. Show that the Frobenius norm is different from the induced matrix-2 norm.

3 BV exercises

5.1, 5.2, 5.4, 5.6, 5.8, 5.9, 6.3, 6.11, 6.17, 6.21, 6.22