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
Diagonalising Matrices / Recurrence Relations Watch
- Thread Starter
Last edited by TheEd; 03-12-2010 at 21:48.
- 03-12-2010 20:56
- 04-12-2010 00:39
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