ALAFF Additional homework

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Subsection 9.5.1 Additional homework

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Homework 9.5.1.1.

Let $\| \cdot \|$ be matrix norm induced by a vector norm $\| \cdot \| \text{.}$ Prove that for any $A \in \Cmxm \text{,}$ the spectral radius, $\rho( A )$ satisfies $\rho( A ) \leq \| A \|\text{.}$

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Some results in linear algebra depend on there existing a consistent matrix norm

$\| \cdot \|$ such that

$\| A \| \lt 1 \text{.}$ The following exercise implies that one can alternatively show that the spectral radius is bounded by one:

$\rho( A ) \lt 1\text{.}$

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Homework 9.5.1.2.

Given a matrix $A \in \Cmxm$ and $\epsilon \gt 0 \text{,}$ there exists a consistent matrix norm $\| \cdot \|$ such that $\| A \| \leq \rho( A ) + \epsilon \text{.}$