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Homework Help: Eigenvector and eigenvalues

  1. Oct 29, 2006 #1
    Hey all,

    I have two matrices A,B which commute than I have to show that these eigenvectors provide a unique classification of the eigenvectors of H?

    From these pairs of eigenvalue is it possible to obtain the eigenvectors?


    I dont quite know how to procede any suggestions?

    Thanks in advance
     
  2. jcsd
  3. Oct 29, 2006 #2

    StatusX

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    What is H?
     
  4. Oct 29, 2006 #3

    HallsofIvy

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    This makes no sense. Obviously, the eigenvectors of A and B will tell you nothing about the eigenvectors of some arbitrary third matrix H. What relationships are there between A, B, and H?

    Yes. Look closely at how A and B are related to H.
     
  5. Oct 29, 2006 #4
    Sorry not H but A the same matrix
     
  6. Oct 29, 2006 #5

    StatusX

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    So you're asking if the eigenvectors of B determine the eigenvectors of A, given that A and B commute? This doesn't sound right, since the identity matrix commutes with everything. You can narrow down the possible eigenvectors of A, but you won't get a "unique classification."
     
  7. Oct 30, 2006 #6
    to see if I understand correctly - let's assume that the matrix A har the eigenvalues {1,2,2} and the matrix B has the eigenvalues {-1,1,1} - then it is possible to construct the eigenvectors of B according to the common unique pairs of A and B( (1,1),(2,1),(2,-1)) giving the following eigenvectors: (1,0,0) , (0,1,1) , (0,-1,1) ?

    And had it not been possible with unique pairs of eigenvalues would the eigenvectors not be orthogonanle?

    Thanks in advance
     
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