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# Diagonalising Matrices / Recurrence Relations Watch

1. For parts (a) and (b) I've found the eigenvalues to be and with corresponding eigenvectors and 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

We can diagonalise by:

letting and so that we have

Now how do I find from here?

EDIT: I see that
2. Assuming you're aiming to get a matrix A to the power n:

Find P such that:
P^-1 A P = D, where D is diagonal. Then:
A = (P D P^-1) and:
A^n = (P D P^-1)(P D P^-1)(P D P^-1)....

The P^-1 P terms cancel and you're left with:
A^n = P D^n P^-1

... and this is much easier to calculate as D^n is trivial

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Updated: December 4, 2010
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