Consider an one-order Linear Recurrence of a vector sequence, such as $${\bf x}_{n+1}={\bf A}{\bf x}_n$$ where ${\bf x}_n \in \mathbb{R}^m (\forall n)$, and ${\bf A} \in \mathbb{R}^{m\times m}$ and ${\bf x}_0$ are constants.

Denote $x_{n,i}$ be the $i$th item of ${\bf x}_n$. My question is for some $i$, whether sequence $\{x_{j,i}\}_{j \in \mathbb{N}_+}$ is always a Linear Recurrence. If it is, can we give an upper bound on the order of the Linear Recurrence? Futhermore, if the upper bound is $k$, is such order always a factor of $k$?