• Support PF! Buy your school textbooks, materials and every day products Here!

Matrix Powers

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

Attachments

Last edited:

Answers and Replies

Dick
Science Advisor
Homework Helper
26,258
618
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.
 
Last edited:

Related Threads for: Matrix Powers

  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
6
Views
1K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
7
Views
698
  • Last Post
Replies
11
Views
822
Replies
4
Views
17K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
2
Views
3K
Top