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Positive Definite

  1. Aug 26, 2008 #1
    URGENT: Can you prove or disprove:

    Let A and B be (complex matrices) positive definite with trace 1.

    Given A < B, (B-A is pos def )


    A^2 < AB (AB-A^2 is pos def)
  2. jcsd
  3. Aug 27, 2008 #2


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    Maybe you can use that
    A B - A^2 = A(B - A)
    and from the result that
  4. Aug 27, 2008 #3
    A or B might not be real symmetric.
  5. Aug 28, 2008 #4
    This is not homework! But I guess this section will get more viewers.
  6. Aug 28, 2008 #5


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    Usually self-adjointness is included in any notion of positivity for complex operators. How are you defining "positive definite" for a complex matrix A?
  7. Aug 29, 2008 #6
    A matrix M such that for all vectors v, <v, Mv> (inner product, the usual one for complex vector spaces) is a real, positive number.
  8. Aug 29, 2008 #7


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    But if <v, Mv> is real for all v, then M is self-adjoint.
  9. Aug 29, 2008 #8
    M is indeed self-adjoint.
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