Exercise 6.10.1

Answers

1.: No. The system $(\begin{matrix} k^{2} & 0 \\ 0 & 1 \end{matrix})$ is well-conditioned but its condition number is $k$ , which can be arbitrarily large.
2.: No. This is the contrapositive statement of the previous question.
3.: Yes. This is the result of Theorem 6.44.
4.: No. The norm of $A$ is a value but the Rayleigh quotient is a function.
5.: No. See the Corollary 1 after Theorem 6.43. For example, the norm of $(\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix})$ is $1$ but the largest eigenvalue of it is $0$ .

jephianlin

2011-06-27 00:00