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Obtain an eigenvector corresponding to each eigenvalue

  1. Dec 5, 2007 #1
    1. The problem statement, all variables and given/known data
    The linear operator T on R^2 has the matrix [4 -5; -4 3]
    relative to the basis { (1,2), (0,1) }

    Find the eigenvalues of T.
    Obtain an eigenvector corresponding to each eigenvalue.


    2. Relevant equations



    3. The attempt at a solution

    I was able to find the eigenvalues (8 and -1) easily enough; however, I have not been able to find the eigenvectors. (I have a feeling it's due to a nonstandard basis being given.)

    AX = 8X
    AX = -1X

    In the case of the first equation I get 4a - 5b = 8a; -4a + 3b = 8b so I would think an eigenvector could be (-5, 4) however my book says it should be (-5,-6). I believe this is because of the basis, but I really don't know what to do with it.
     
  2. jcsd
  3. Dec 5, 2007 #2

    mjsd

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    are you sure it is (-5,-6)? or did u write down the basis correctly? {(1,2),(0,1)}

    EDIT: ok i think it is correct, my mistakes
     
    Last edited: Dec 5, 2007
  4. Dec 5, 2007 #3
    Yep, according to my book the answer is (-5,-6) and the basis is correct, is the book wrong?
     
  5. Dec 5, 2007 #4

    mjsd

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    is the eigenvector corresponding to evalue 1 is (1,3) according to book?
     
  6. Dec 5, 2007 #5
    yep.. I can see that (-5,4) -> -5(1,2) + 4(0,1) = (-5,-6)
    is (-5,4) a coordinate matrix?
     
    Last edited: Dec 5, 2007
  7. Dec 5, 2007 #6

    mjsd

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    yes... but I can't pinpoint the cause of the problem in your method just yet...
     
  8. Dec 5, 2007 #7

    mjsd

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    by the way, i believe the answers in the book is written in terms of the standard basis {(1,0), (0,1)}
     
  9. Dec 5, 2007 #8

    mjsd

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    ok, previously I made two mistakes that confused myself
    note: -5 (1,2) +4 (0,1) = (-5,-6)
    and 1 (1,2) +1 (0,1) = (1,3)
     
  10. Dec 5, 2007 #9
    alright i think i can see it now, thanks
     
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