TITLE: Converting recursive equations into matrices
QUESTION [29 upvotes]: How do we convert recursive equations into matrix forms?
For instance, consider this recursive equation(Fibonacci Series): $$F_n = F_{n-1} + F_{n-2}$$
And it comes out to be that the following that 
$$\begin{bmatrix}1&1\\1&0\end{bmatrix}^n = \begin{bmatrix}F_{n+1}&F_{n}\\F_{n}&F_{n-1}\end{bmatrix}$$
Can please anyone tell me how do we derive such a base matrix for recursive equations? How can we determine the order of the matrix for the recursive equation, as well as the elements of the matrix?

REPLY [9 votes]: Imagine you have recursive relation, for example $a_n=\alpha a_{n-1}+ \beta a_{n-2}$. You try to find such a matrix $A$, that:
$$A \begin{bmatrix}a_{n} \\ a_{n-1}\end{bmatrix}=\begin{bmatrix}a_{n+1} \\ a_{n}\end{bmatrix}$$
It will be $2 \times 2$ matrix. Let $A=\begin{bmatrix}a && b \\ c && d\end{bmatrix}$, so:
$$A\begin{bmatrix}a_{n} \\ a_{n-1}\end{bmatrix}=\begin{bmatrix}a \cdot a_{n}+b \cdot a_{n-1} \\ c \cdot a_{n}+d \cdot a_{n-1}\end{bmatrix}$$
You want have:
$$a \cdot a_{n}+b \cdot a_{n-1}=a_{n+1}=\alpha a_{n}+ \beta a_{n-1}$$
$$c \cdot a_{n}+d \cdot a_{n-1}=a_{n}$$
If you solve this system of equation you get:
$$A=\begin{bmatrix}\alpha && \beta \\ 1 && 0\end{bmatrix}$$
You can you this method for other recursive relation, for example $a_n=\alpha a_{n-1}+ \beta a_{n-2}+\gamma a_{n-3}$ or any numbers of components.