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Spectral theorem for self-adjoint linear transformations

  1. Feb 22, 2010 #1
    Let P,Q be self-adjoint linear transformations from V to V, Q is also positive-definite. Deduce that there exist scalars λ1 , . . . , λn and linearly independent vectors e1 , . . . , en in V such that, for i, j = 1, 2, . . . , n:
    (i) P ei = λi Qei ;
    (ii) <P ei , ej > = δi j λi ;
    (iii) <Qei , ej > = δi j .
    I could use the spectral theorem to find an orthonormal basis ei for P and Q separately, but how can I connect them together? Could anyone give me some hint? Any help is greatly appreciated!
  2. jcsd
  3. Feb 23, 2010 #2
    Let Ei be the linear transformations that project into the subspace spanned by ei. Then (i) is essentially saying that

    [tex]P = \sum \lambda_i QE_i = Q \sum \lambda_i E_i[/tex].

    I hope I'm not giving away too much here.
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