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Ted123
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Homework Statement
[PLAIN]http://img530.imageshack.us/img530/6672/linn.jpg
The Attempt at a Solution
For parts (a) and (b) I've found the eigenvalues to be [itex]-\frac{1}{3}[/itex] and [itex]-1[/itex] with corresponding eigenvectors [itex]\begin{bmatrix} -1 \\ 3 \end{bmatrix}[/itex] and [itex]\begin{bmatrix} -1 \\ 1 \end{bmatrix}[/itex] respectively.
Now for part (c) I know there is a way of solving this by diagonalising matrices but I can't remember the method.
The recurrence relation can be written as [itex]\begin{bmatrix} a_n \\ a_{n-1} \end{bmatrix} = \begin{bmatrix} -\frac{4}{3} & -\frac{1}{3} \\ 1 & 0 \end{bmatrix} \begin{bmatrix} a_{n-1} \\ a_{n-2} \end{bmatrix}[/itex]
We can diagonalise [itex]A = \begin{bmatrix} -\frac{4}{3} & -\frac{1}{3} \\ 1 & 0 \end{bmatrix}[/itex] by:
letting [itex]D = \begin{bmatrix} -\frac{1}{3} & 0 \\ 0 & -1 \end{bmatrix}[/itex] and [itex]P = \begin{bmatrix} -1 & -1 \\ 3 & 1 \end{bmatrix}[/itex] so that we have [itex]A= PDP^{-1}[/itex]
Now how do I find [itex]a_n[/itex] from here?
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