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Homework Help: 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


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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


    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


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  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


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    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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