Chapter 4.3, Problem 8E

Suppose that $A=S\wedge S^{-1}$ , where $\wedge$ is a diagonal matrix with diagonal elements ${\lambda }_1,{\lambda }_2,\mathrm{...},{\lambda }_n$ .
a) Show that $A{s}_i={\lambda }_i{s}_i,i=1,\mathrm{...},n$ .
(b) Show that if $x={\alpha }_1{s}_1+{\alpha }_2{s}_2+\mathrm{...}+{\alpha }_n{s}_n$ , then $A^{k}x={\alpha }_1{\lambda }_1^k{s}_1+{\alpha }_2{\lambda }_2^k{s}_2+\mathrm{...}+{\alpha }_n{\lambda }_n^k{s}_n$ .
(c) Suppose that $\left|{\lambda }_i\right|<1$ for $i=1,\mathrm{...},n$ . What happens to $A^{k}x$ as $k\to \infty$ ? Explain.