Let .

Find formulas for the entries of , where is a positive integer.

.

I'm not sure how to go about this. Could someone give push me in the right direction?

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- November 4th 2007, 09:45 AMThomasEigen Diagonalization
Let .

Find formulas for the entries of , where is a positive integer.

.

I'm not sure how to go about this. Could someone give push me in the right direction? - November 4th 2007, 06:42 PMtopsquark
Probably the simplest thing to do would be to calculate what multiplying M by an arbitrary 2 x 2 matrix would give:

Now you can set up a recursion: for a = 5, b = 2, c = -1, and d = 2. This generates the next set of a, b, c, and d.

So , , etc.

Now solve the system of recursions. (That step is beyond me, I'm afraid!)

-Dan - November 7th 2007, 08:14 PMThePerfectHacker
That is not how to do it topsquark. You need to diagnolize the matrix.

Meaning,

.

Where is some invertible matrix and is a diagnol matrix. Once you do that the above equation is easy to handle.

Bring everything to power of ,

Thus,

And,

Where can be found from its eigenvectors and from its eigenvalues.

(Note, computing a power of a diagnol is easy. Just raise each diagnol entry to that exponent).