$\def\vct{\overrightarrow} \def\T{^{\mathrm{T}}}$Suppose there is a system of recursive equations as$$
\vct{x}_{n + 1} = A \vct{x}_n, \quad \forall n \in \mathbb{N}
$$
where $A \in M_{m \times m}(\mathbb{R})$ and $\vct{x}_n = (x_{n, 1}, \cdots, x_{n, m})\T$ for all $n$. By induction on $n$, it is easy to see that $\vct{x}_n = A^n \vct{x}_0 \ (\forall n \in \mathbb{N})$, where $A^0 := I$.
Now define $ε_k = (0, \cdots, 0, 1, 0, \cdots, 0)\T \in \mathbb{R}^m$ for $1 \leqslant k \leqslant m$, where the $k$-th component of $ε_k$ is $1$, and the translation operator $B$ as $B x_{n, k} = x_{n + 1, k}$ for all $1 \leqslant k \leqslant m$ and $n$. Suppose$$
φ(λ) = λ^l + \sum_{j = 0}^{l - 1} c_j λ^j
$$
is the minimal polynomial of $A$ over $\mathbb{R}$ (or any polynomial $φ(λ) \in \mathbb{R}[λ]$ such that $φ(A) = O$). Note that for a fixed $1 \leqslant k \leqslant m$, there is $x_{n, k} = ε_k\T \vct{x}_n$ for all $n$, therefore\begin{align*}
φ(B) x_{n, k} &= x_{n + l, k} + \sum_{j = 0}^{l - 1} c_j x_{n + j, k}\\
&= ε_k\T \left( A^l + \sum_{j = 0}^{l - 1} c_j A^j \right) \vct{x}_0\\
&= ε_k\T φ(A) \vct{x}_0 = 0
\end{align*}
for all $n$. Hence the recursive formula for $\{x_{n, k}\}_{n \geqslant 0}$ is$$
x_{n + l, k} = -\sum_{j = 0}^{l - 1} c_j x_{n + j, k}. \quad \forall n \in \mathbb{N}
$$
Example 1. For the recursive system$$
\begin{cases}
a_{n + 1} = a_n + 2b_n\\
b_{n + 1} = 2a_n + 2b_n
\end{cases},
$$
there is$$
A = \begin{pmatrix}
1 & 2\\
2 & 2
\end{pmatrix}.
$$
Note that $A^2 - 3A - 2 = O$, thus $φ(λ)$ can be taken as $φ(λ) = λ^2 - 3λ - 2$. Therefore the recursive formulae for $\{a_n\}$ and $\{b_n\}$ are\begin{align*}
a_{n + 2} &= 3a_{n + 1} + 2a_n,\\
b_{n + 2} &= 3b_{n + 1} + 2b_n.
\end{align*}
Example 2. For the recursive system$$
\begin{cases}
a_{n + 1} = a_n + b_n + c_n\\
b_{n + 1} = a_n + b_n + c_n\\
c_{n + 1} = a_n + b_n + c_n + d_n\\
d_{n + 1} = c_n + d_n
\end{cases},
$$
there is$$
A = \begin{pmatrix}
1 & 1 & 1 & 0\\
1 & 1 & 1 & 0\\
1 & 1 & 1 & 1\\
0 & 0 & 1 & 1
\end{pmatrix}.
$$
Take $φ(λ)$ as the characteristic polynomial of $A$, i.e.$$
φ(λ) = |λI - A| = λ^4 - 4λ^3 + 2λ^2 + 2λ.
$$
By the Cayley–Hamilton theorem, $φ(A) = O$. Therefore the recursive formulae for $\{a_n\}$, $\{b_n\}$, $\{c_n\}$ and $\{d_n\}$ are\begin{align*}
a_{n + 4} &= 4a_{n + 3} - 2a_{n + 2} - 2a_{n + 1}\\
b_{n + 4} &= 4b_{n + 3} - 2b_{n + 2} - 2b_{n + 1}\\
c_{n + 4} &= 4c_{n + 3} - 2c_{n + 2} - 2c_{n + 1}\\
d_{n + 4} &= 4d_{n + 3} - 2d_{n + 2} - 2d_{n + 1}.
\end{align*}
