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Higher Power of a square Matrix

  1. Oct 22, 2011 #1
    1. The problem statement, all variables and given/known data

    Given the matrix A=
    -1/5 7/5
    -3/5 -4/5

    find A43.





    3. The attempt at a solution
    It's obvious that I can't go and actually compute A43 so there must be a more elegant way of doing this. The only notes I have on the subject is Ak=P-1DkP, where D is a diagonal matrix. However, I have no clue what P is supposed to to, nor why this would work in the first place.
     
  2. jcsd
  3. Oct 22, 2011 #2

    phyzguy

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    Science Advisor

    You're on the right track. If you've studied eigenvalues and eigenvectors, you should be able to calculate D and P.
     
  4. Oct 22, 2011 #3
    I've never heard of eigenvalues or eigenvectors. I'm in a college level Linear Algebra I class, so I don't know if I should or not.
     
  5. Oct 22, 2011 #4

    Mark44

    Staff: Mentor

    You need to have some understanding of eigenvalues and eigenvectors to be able to diagonalize a matrix. In your formula, the columns of matrix P are the eigenvectors of matrix A, and P-1 is the inverse of P. Matrix D is a diagonal matrix whose entries are the eigenvalues of A.

    If you're expected to work a problem like this, there must be similar problems in your textbook, and some presentation of these ideas must have been given in class.
     
  6. Oct 22, 2011 #5

    phyzguy

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    Science Advisor

  7. Oct 22, 2011 #6
    Ah ok, it's quite possible that the prof mentioned it but I missed it. Thanks for the link, I'll put it to good use.
     
  8. Oct 22, 2011 #7

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Also, the eigenvalues for this particular matrix are complex numbers. That's going to make calculating the 43 power even more complicated. Fortunately, they both have modulus 1.
     
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