Higher Power of a square Matrix

  • Thread starter Guidenable
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  • #1
Guidenable
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Homework Statement



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

find A43.





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.
 

Answers and Replies

  • #2
phyzguy
Science Advisor
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You're on the right track. If you've studied eigenvalues and eigenvectors, you should be able to calculate D and P.
 
  • #3
Guidenable
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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.
 
  • #4
36,483
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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
Guidenable
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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.
 
  • #7
HallsofIvy
Science Advisor
Homework Helper
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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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