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I had a recurrence equations: $\left\{\begin{matrix} a_{n}= a_{n-1}+b_{n-1}\\ b_{n-1}= a_{n-4}+a_{n-5} \end{matrix}\right. (1)$
How can one get the generating matrix so that it can be solved and find the characteristic polynomial of it?
The Fibonacci sequence $f_{0}, f_{1},....$ is defined by the recurrence relation. $ f_{n+1}=f_{n}+f_{n-1} $
then we have: $\begin{pmatrix} 0 & 1\\ 1 & 1 \end{pmatrix}\begin{pmatrix} f_{n-1}\\ f_{n} \end{pmatrix}=\begin{pmatrix} f_{n}\\ f_{n+1} \end{pmatrix}$
$A=\begin{pmatrix} 0 &1 \\ 1 &1 \end{pmatrix}$ is the corresponding generating matrix and the characteristic polynomial of it is $ x^2-x-1.$
So how to get the generating matrix for any given recurrence equations?
P/s: I solved (1) by substitution and got the characteristic polynomial that is:
$x^5-x^4-x-1$.
Thanks.

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