Exercise 7.2.19

Answers

1.: It comes from some direct computation. Multiplying $N$ at right means moving all the columns to their right columns.
2.: Use Exercise 7.2.18(c). Now the matrix $M$ is the matrix $N$ in Exercise 7.2.19(a).
3.: If $| λ | < 1$ , then the limit tends to a zero matrix. If $λ = 1$ amd $m = 1$ , then the limit tends to the identity matrix of dimension $1$ . Conversely, if $| λ | \geq 1$ but $λ \neq 1$ , then the diagonal entries will not converge. If $λ = 1$ but $m > 1$ , the $12$ -entry will diverge.
4.: Observe the fact that if $J$ is a Jordan form consisting of several Jordan blocks $J_{i}$ , then $J^{r} = ⊕_{i} J_{i}^{r}$ . So the $\lim_{m \to \infty} J^{m}$ exsist if and only if $\lim_{m \to \infty} J_{i}^{m}$ exists for all $i$ . On the other hand, if $A$ is a square matrix with complex entries, then it has the Jordan canonical form $J = Q^{- 1} AQ$ for some $Q$ . This means that $\lim_{m \to \infty} A^{m}$ exists if and only if $\lim_{m \to \infty} J^{m}$ exists. So Theorem 5.13 now comes from the result in Exercise 7.2.19(c).

jephianlin

2011-06-27 00:00