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?
Best Answer
If $F_n$ is a linear function of $F_{n-1},F_{n-2},\dots,F_{n-k}$ with constant coefficients, then you'll need a $k \times k$ matrix to represent the recurrence. Intuitively this is because the "state" of the recurrence is the previous $k$ values: you need exactly those values to compute the next one.
As for actually finding the matrix, you need to find $A$ such that (in the case of a second order recurrence):
$$\begin{bmatrix} F_n \\ F_{n-1} \end{bmatrix} = A \begin{bmatrix} F_{n-1} \\ F_{n-2} \end{bmatrix}.$$
The second row of $A$ is clear: $F_{n-1} = F_{n-1}$, so the second row should be $\begin{bmatrix} 1 & 0 \end{bmatrix}$. The recurrence itself lives in the first row; in the Fibonacci case we have $F_n = F_{n-1} + F_{n-2}$ so the first row is $\begin{bmatrix} 1 & 1 \end{bmatrix}.$