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Splitting sigmas

  1. Nov 12, 2012 #1
    1. The problem statement, all variables and given/known data Prove the following statement, where A and B are nxn matrices.

    [tex] Tr(AB) = Tr(BA) [/tex]

    2. Relevant equations

    3. The attempt at a solution
    Using some manipulations, I arrived at
    [tex] \sum^{n}_{p=1}\sum^{n}_{k=1}A_{pk}B_{kp} = \sum^{n}_{p=1}\sum^{n}_{k=1}B_{pk}A_{kp} [/tex]

    If I can prove the above, I am done, but I have never worked with double sigma notations before, so any advice folks?

  2. jcsd
  3. Nov 12, 2012 #2


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    Science Advisor
    Homework Helper

    k and p are what are called 'dummy indices'. You can change k to any other letter, like q and you still have a good formula for Tr. Just interchange k and p.
  4. Nov 13, 2012 #3
    Ah, ingenious! Thanks!

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