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Homework Help: Linear algebra of operators

  1. Sep 17, 2006 #1
    If you take the outer product of two orthogonal 4-dimensional vectors you get a 4*4 matrix. I wonder what you generally can say about this matrix eigenvectors? For example: If 2,3 or 4 of the eigenvalues are zero is it always true that the matrix have at least 2 linear dependent eigenvectors? Maybe this is true regardless of the values of the eigenvalues?

    The physical problem is about the operator |a><b| where |a> and |b> are orthogonal to each other and the characteristic polynom is x^4. I have the values for this operator in a basis and it can not be diagonalized.
    Is this maybe a general property when |a> and |b> are orthogonal? Or when a couple of the eigenvalues are zero?
    If so what is the physics behind this?

    Any ideas?
  2. jcsd
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