###### Exercise 9

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*}

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

Find the eigenvalues and associated eigenvectors of \(A\text{.}\)

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

Write \(\xvec_{0}\) as a linear combination of eigenvectors of \(A\text{.}\)

Write \(\xvec_n\) as a linear combination of eigenvectors of \(A\text{.}\)

What happens to \(G_n\) as \(n\) becomes very large?