Periodic sequence

It's easy enough to check that the first time we get $\bold{v}(k)=\bold{v}(0)$ is when $k=\boxed{18}$ . It's a bit more interesting to work out why.

What kind of geometric transformation in $3$ D can be periodic? It must be the case that $A$ is either a rotation or a rotation with a reflection.

We can verify that $|A|=1$ ; so it is a rotation.

The trace of a rotation matrix depends only on the angle of rotation (ie not the axis of rotation). This is because the trace is invariant under change of basis; using the property that $\text{tr}(UV)=\text{tr}(VU)$ , we have $\text{tr}(P^{-1}AP) =\text{tr}(P^{-1}(AP))=\text{tr}((AP)P^{-1})=\text{tr}(A)$

Knowing that $\begin{pmatrix} 1 & 0 & 0\\ 0 & \cos\theta & -\sin\theta\\ 0 & \sin\theta & \cos\theta \end{pmatrix}$

is a rotation matrix, we find $\text{tr}(A)=1+2\cos\theta$

for any $3$ D rotation matrix. In this case, $\text{tr}(A)=2.879\ldots$

and solving we find $\theta=0.349\ldots \approx \frac{\pi}{9}$

so again we find the period is $\boxed{18}$ .