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 $k$?



For an one-order Linear Recurrence of a vector sequence, does the corresponding item follow a Linear Recurrence?
each of the $m$ component sequences of your $X$ satisfies a linear recurrence of degree $m$ with characteristic polynomial given by Cayley-Hamilton. The point, I suppose, is that 
$$ X\_{n+j} = A^j X\_n $$


