Recursive formula to 3x3 matrix

Question

I was given a recursive formula and I need to convert it into a $3\times3$ matrix. What is a general formula to do this?

My recursion is in the form: $$R_{n+2} = 4R_{n+1} + 5R_n + 2R_{n-1}$$

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Just want a general equation to help me solve this!! Thanks

Answer 1

The usual approach is to express the new term and last few terms as linear combinations of the old terms:

$\begin{array}{lrrrr}R_{n+2} &=& 4R_{n+1} &+ 5R_n& + 2R_{n-1}\\ R_{n+1}&=& R_{n+1}\\ R_n&=&&R_n\end{array}$

Now just turn that into a matrix :-).

Answer 2

Let $B_n=[R_{n+2}\;R_{n+1}\;R_{n}]^\top$. $$ \begin{bmatrix} 4&5&2\\ 1&0&0\\ 0&1&0 \end{bmatrix} \begin{bmatrix} R_{n+1}\\ R_n\\ R_{n-1} \end{bmatrix}= \begin{bmatrix} R_{n+2}\\ R_{n+1}\\ R_{n} \end{bmatrix} $$ or, $AB_{n-1}=B_n\implies B_n=A^nB_0=JD^nJ^{-1}B_0$ assuming $A$ can be diagonalized and $A=JDJ^{-1}$ where $D$ is a diagonal matrix.