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I am working on this problem with a third order equation: $t_{n+2}=3t_{n+1}+t_n-3t_{n-1}$

I am asking to change this to a matrix and honestly I can not quite understand How the change works

Can someone help me a bit?

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    it is very hard to understand your equation – 2017-02-07
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    for all n part should be under t I don't know how to present that sorry – 2017-02-07
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    do you mean $t_n+2=3t_n+1+t_n-3t_n-1$? – 2017-02-07
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    (n+2) (n+1) (n) (n-1) all together – 2017-02-07
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    ok, I got it. Use this to improve your latex: http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – 2017-02-07
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    do you want solve the recurrence or necessarily use a matrix to do that? – 2017-02-07
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    I need to solve it using 3x3 matrix recursion – 2017-02-07

1 Answers 1

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You can write:

$$x_{n+1}=\begin{pmatrix}t_{n+1}\\ t_{n}\\ t_{n-1} \end{pmatrix}$$

and we will try to find a matrix $M$ such that $x_{n+1}=M\cdot x_{n}$. Looking to the recurrence we get:

$$\begin{pmatrix}t_{n+2}\\ t_{n+1}\\ t_{n} \end{pmatrix}=\begin{pmatrix} 3&1&-3\\ 1&0&0\\ 0&1&0 \end{pmatrix}\cdot \begin{pmatrix}t_{n+1}\\ t_{n}\\ t_{n-1} \end{pmatrix}$$

Now you need a initial condition such that $t_0,t_1,t_2$ and then you have $x_0$.

After that you can use that $x_n=M^n\cdot x_0$

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    Thank u so much that helps a lot!! – 2017-02-07
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    you are very welcome! – 2017-02-07