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Subsection 9.5.1 Additional homework

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 \(\rho( A ) \leq \| A \|\text{.}\)

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

Homework 9.5.1.2.

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