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Proof problem(Linear Algebra- Eigenvalues/Eigenvectors)

  1. Dec 11, 2011 #1
    1. The problem statement, all variables and given/known data
    True/False
    The geometric multiplicity of an eigenvalue of a symmetric matrix necessarily equals to its algebric multiplicity.

    2. Relevant equations


    3. The attempt at a solution
    True.
    If a matrix is symmetric, then the matrix is diagonalizable. Since the matrix is diagonalizable, there must be eigenvectors correspond to each eigenvalues.


    So, I did the proof, but I'm not so sure if it sounds right. I think there could be something more tricky or missing. Would you guys check if this sounds right to you?
     
  2. jcsd
  3. Dec 11, 2011 #2

    Dick

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    Sound right to me.
     
  4. Dec 11, 2011 #3
    Thanks for checking. Just quick checking tho,
    'A matrix is symmetric if and only if the matrix is diagonalizable.'

    Is this a right statement?

    or 'orthogonally diagonalizable'
     
  5. Dec 11, 2011 #4

    Dick

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    No, it's not iff. Is the matrix [[1,1],[0,0]] diagonalizable? Is it symmetric?
     
  6. Dec 11, 2011 #5
    oh, thanks!
     
  7. Dec 11, 2011 #6

    Dick

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    Well, it is true that "orthogonally diagonalizable" iff symmetric.
     
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