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Show that the eigenvalues of the overlap matrix are positive

  1. Feb 17, 2008 #1
    1. The problem statement, all variables and given/known data
    The task is to show that the eigenvalues of overlap matrix [tex]\tilde S[/tex] are positive.

    2. Relevant equations
    The overlap matrix is defined as [tex] (\tilde S)_{nm} = \langle \xi_n \vert \xi_m \rangle [/tex], with [tex]\xi_k[/tex] being the base vectors of the wavefunction. http://en.wikipedia.org/wiki/Overlap_matrix

    3. The attempt at a solution
    I've tried to show that the eigenvalues are positive by showing that [tex]\tilde S[/tex] is positive definite. Both with the condition [tex]\vec x^{\ast} \tilde S \vec x > 0[/tex] and the condition that all the 'sub-determinants' are larger than zero. But I don't get it right, please help.
  2. jcsd
  3. Feb 18, 2008 #2
    In a naively way i would say: It is just an integral of a probabilty distribution---> it is positive by definition!!

    More thecnically: the diagonal members are for sure positive and what about the off diagonal?

    Hint: use the fact that [tex]S_{jk}=\overline{S}_{kj}[/tex]
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