Matrix Powers - General Expression for Mk to the Nth Power

[masterprimus]

This problem first appeared on another thread under Statistics and probability. I found it when I got the same problem, which is as follows

Consider matrices in the form (k+1 k-1)
(k-1 k+1)

We will call this matrix Mk, find a general expression for Mk to the nth power in terms of k and n.


I tried several different matrices of this form, the general expression i came up with is attached in the thumbnail, but I still tried making it somewhat clear with latex.

M$$^{N}_{K}$$ = 2$$^{n-1}$$$$\left( [(k+1) +(k - 1)\sum^{n}_{x=1} k^{x}] [(k-1) +(k - 1)\sum^{n}_{x=1} k^{x}] \right)$$
$$\left([(k - 1) +(k - 1)\sum^{n}_{x=1} k^{x}] [(k + 1) +(k - 1)\sum^{n}_{x=1} k^{x}] \right)$$

 

[Dick]

I don't think that's quite right, if that was your question. But there is a systematic way to do this. M can be diagonalized since it's symmetric. Find the matrix which diagonalizes it so M=S^(-1).D.S where D is diagonal. Then M^n=S^(-1).D^n.S.