Exercise 9

Gil Strang defines the Gibonacci numbers $G_n$ as follows. We begin with $G_0 = 0$ and $G_1=1$. A subsequent Gibonacci number is the average of the two previous; that is, $G_{n+2} = \frac12(G_{n}+G_{n+1})$. We then have $$\begin{aligned} G_{n+2} & {}={} \frac12 G_{n} + \frac 12 G_{n+1} \\ G_{n+1} & {}={} G_{n+1}\text{.} \\ \end{aligned}$$

1. If $\xvec_n=\twovec{G_{n+1}}{G_n}$, find the matrix $A$ such that $\xvec_{n+1} = A\xvec_n$.

2. Find the eigenvalues and associated eigenvectors of $A$.

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 $A$.

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

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