Exercise 5.3.1

Answers

1.: Yes. This is the result of Theorem 5.12.
2.: Yes. This is the result of Theorem 5.13.
3.: No. It still needs the condition that each entry is nonnegative.
4.: No. It’s the sum of each column. The matrix $(\begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix})$ is a counterexample.
5.: Yes. See the Corollary after Theorem 5.15, although there’s no proof.
6.: Yes. See the Corollary 1 after Theorem 5.16.
7.: Yes. Look Theorem 5.17.
8.: No. The matrix $(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})$ has eigenvector $(1, - 1)$ corresponding the eigenvalue $- 1$ .
9.: No. The matrix $A = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})$ has the property that $A^{2} = I$ . So the sequence would be $A, I, A, I, \dots$ , which does not converge.
10.: Yes. This is Theorem 5.20.

jephianlin

2011-06-27 00:00