##### Exercise9

Gil Strang defines the Gibonacci numbers $$G_n$$ as follows. We begin with $$G_0 = 0$$ and $$G_1=1\text{.}$$ A subsequent Gibonacci number is the average of the two previous; that is, $$G_{n+2} = \frac12(G_{n}+G_{n+1})\text{.}$$ We then have

\begin{equation*} \begin{aligned} G_{n+2} \amp {}={} \frac12 G_{n} + \frac 12 G_{n+1} \\ G_{n+1} \amp {}={} G_{n+1}\text{.} \\ \end{aligned} \end{equation*}
1. If $$\xvec_n=\twovec{G_{n+1}}{G_n}\text{,}$$ find the matrix $$A$$ such that $$\xvec_{n+1} = A\xvec_n\text{.}$$

2. Find the eigenvalues and associated eigenvectors of $$A\text{.}$$

3. Explain why this dynamical system does not neatly fit into one of the six types that we saw in this section.

4. Write $$\xvec_{0}$$ as a linear combination of eigenvectors of $$A\text{.}$$

5. Write $$\xvec_n$$ as a linear combination of eigenvectors of $$A\text{.}$$

6. What happens to $$G_n$$ as $$n$$ becomes very large?

in-context