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How to prove trace(A.A*) is positive

  1. Dec 14, 2006 #1
    Hello

    I'd like to know how to prove that the trace of A.A* is positive.
    I don't really know how to handle the imaginary part of it. If A has any complex number in it, is it possible to get traces like 10-2i? If yes, do I consider it as a positive number or negative?:zzz:

    Thanks in advance
     
  2. jcsd
  3. Dec 14, 2006 #2

    morphism

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    First of all, the diagonal entries of AA* are real. You can't really compare two compex numbers like that as there is no order on C.

    Now, what does the (i,j)-th entry of AA* look like? What about the (i,i)-th entry?

    (Side note: tr(AA*) isn't always positive - it can be zero. So a better thing would be to say that it's nonnegative.)
     
  4. Dec 15, 2006 #3
    My sketchy knowledge about linear algebra tells me that you would have to relate the nature of the singular values of AA* to its trace being positive.
     
  5. Dec 15, 2006 #4

    matt grime

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    That would not be a very easy way of doing this question. The trace of (AA*) is

    [tex] \sum_{i,j} A_{ij}(A^*)_{ji}[/tex]

    What is the definition of A*?
     
  6. Dec 15, 2006 #5
    Ah yes, thank you for the note.
     
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