For an one-order Linear Recurrence of a vector sequence, does the corresponding item follow a Linear Recurrence? [closed]

Question

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 $k$?

Answer 1

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 $$ X_{n+j} = A^j X_n $$