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Eigenvalue and and eigenvector

  1. Jun 5, 2014 #1
    Hi, I have a problem with the calculation of the eigenvalue of a matrix. That matrix is an N x N matrix which can be written as:

    ##M^{ab} = A\delta^{ab} + B \phi^a \phi^b##

    where ##\delta^{ab}## is the identity matrix and the ##\phi## is a column vector. The paper I'm studying says that the eigenvalue of this matrix are:

    A with molteplicity 1

    ##A + \phi^2 B## with molteplicity N-1

    but I can't understand why! Can anyone help me?
     
    Last edited: Jun 5, 2014
  2. jcsd
  3. Jun 5, 2014 #2

    adjacent

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    Gold Member

    You should put ## around the latex code to render it :smile:
     
  4. Jun 5, 2014 #3

    Thank you, now it'right
     
  5. Jun 5, 2014 #4

    AlephZero

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    Science Advisor
    Homework Helper

    I think it should be
    A with multiplicity N-1
    ##A + \phi^2 B## with multiplicity 1.

    First find the eigenvalues of the rank 1 matrix ##B\phi^a\phi^b##.
    Then think about what happens when you add ##A\delta^{ab}##, which is A times the identity matrix.
     
  6. Jun 11, 2014 #5
    yes, you're right, it was A with multeplicity N-1 and the other with multeplicity 1. I tried to use your suggestion and I solved it! thank you very much!
     
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