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Matrix Powers

  1. Mar 17, 2008 #1
    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]
     

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    Last edited: Mar 17, 2008
  2. jcsd
  3. Mar 17, 2008 #2

    Dick

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    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: Mar 17, 2008
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