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Eigenvalue - geometric multiplicity proof

  1. Sep 30, 2009 #1
    1. The problem statement, all variables and given/known data

    Given matrix A:

    a 1 1 ... 1
    1 a 1 ... 1
    1 1 a ... 1
    .. . .. ... 1
    1 1 1 ... a

    Show there is an eigenvalue of A whose geometric multiplicity is n-1. Express its value in terms of a.

    2. Relevant equations

    general eigenvalue/vector equations

    3. The attempt at a solution

    My problem is I'm not sure how to start it off.
    I can state A is square, symmetric and hermitian so I know it has to do with one or more of those. I tried going through using a determinant but it didn't seem to work nicely, I have a feeling that it might have to do with a property of symmetric matrices but am not sure how to go about the proof (or which property)
     
  2. jcsd
  3. Sep 30, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Well, first, what is the eigenvalue in question? To do that, of course, you will need to find the characteristic polynomial. I recommend starting with "1 by 1", "2 by 2", and "3 by 3" matrices to see if you can find a pattern. Do the same thing to find the eigenvectors corresponding to that eigenvalue.
     
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