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Homework Help: Orthogonal transformation

  1. Mar 26, 2008 #1
    1. The problem statement, all variables and given/known data
    My statistical mechanics book says that if M is an real, symmetric n by n matrix, and U is the matrix of its eigenvectors as column vectors, then the transformation UMU^{-1} preserves the trace of M. Is that true? If so, is it obvious? If it is true but not obvious, how do you prove it?

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Mar 26, 2008 #2


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    It's true and obvious. And U doesn't need to be unitary. Use the cyclic property of trace. Tr(ABC)=Tr(CAB).
  4. Mar 27, 2008 #3
    Why is that true?

    EDIT: never mind http://en.wikipedia.org/wiki/Trace_(linear_algebra [Broken])
    Last edited by a moderator: May 3, 2017
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