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Finding a vector A from given eigen values and eigenvectors

  1. Oct 24, 2012 #1
    1. The problem statement, all variables and given/known data

    A matrix A has eigenvectors [2,1] [1,-1]
    and eigenvalues 2 , -3 respectively.

    Determine Ab for the vector b = [1,1].


    2. Relevant equations



    3. The attempt at a solution

    First I put be as a combination of the two eigenvectors
    ie

    2/3[2,1] -1/3[1,-1] = b

    so A(2/3[2,1] -1/3[1,-1]) = Ab

    but not sure what to do from this point as the sltn says it went from this

    A(2/3[2,1] -1/3[1,-1]) = (2[2,1] -1/3[1,-1]) but im not sure how??
     
  2. jcsd
  3. Oct 24, 2012 #2

    tiny-tim

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    hi sg001! :smile:
    no, that can't be right :redface:

    if the second eigenvalue is -3, that factor must be -3, not -1/3
     
  4. Oct 24, 2012 #3
    hmm

    they have the answer of Ab = 1/3[11,1] ???

    is this correct?
     
  5. Oct 24, 2012 #4

    tiny-tim

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    you mean 1/3[11,7] ?

    then the question must be wrong, the eigenvalue must be -1/3
     
  6. Oct 24, 2012 #5
    no, thats the sltn and the exact question they give,,,

    so how would you approach this type of question then..

    rewrite b as a liner combination of the given eigenvalues... then how do you solve for A from here...

    2/3[2,1] -1/3[1,-1] = b

    just so I know if it comes up in my test.

    Thanks for the help.
     
  7. Oct 24, 2012 #6

    tiny-tim

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    hi sg001! :smile:

    you want to find A[1,1].

    start with [1,1] = 2/3[2,1] -1/3[1,-1]

    so A[1,1] = 2/3 A[2,1] -1/3 A[1,-1]

    = 2/3 2[2,1] -1/3 -3[1,-1]


    = [11/3,1/3] …

    oh that is right!

    (i shouldn't have tried doing it in my head :redface:)

    the important step is the bit in bold :wink:
     
  8. Oct 24, 2012 #7
    cool thanks
     
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