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Index Notation question

  1. Sep 21, 2015 #1
    I am having trouble showing that ##(A_{ik}B_{kj})_{mm} = (A_{ki}B_{jk})_{mm}##


    Wouldn't the right side end up having a different outcome? Or can we assume its symmetric?
     
  2. jcsd
  3. Sep 21, 2015 #2

    Mentz114

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    It is true for the anti-symmetric part also because the sign change cancels.
     
  4. Sep 21, 2015 #3
    anti-symmetric?
     
  5. Sep 21, 2015 #4

    Mentz114

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    Yes, I think so. A and B can be decomposed into symmetric and anti-symmetric parts.
    For the antisymmetric part
    ##A_{ik}B_{kj}=(-A_{ki})(-B_{jk})##

    [Edit]

    Is there a 'raised' index here ?

    ##{A_i}^k B_{kj}##
     
  6. Sep 22, 2015 #5

    gill1109

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    It would help if you would explain your notation.

    I suppose that you are using Einstein summation convention. And that when you write (A_{ik}B_{kj}) you mean, the matrix whose i,j element is sum_k (A_{ik}B_{kj}). In other words, the ij element of the matrix AB. Now you want to sum over m the m,m elements of that matrix, so you are talking about trace(AB).

    Similarly on the right hand side, replace i and j both by m and m, sum over m and sum over k, and you see that what you have written is just trace(AB).
     
  7. Sep 23, 2015 #6

    stevendaryl

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    Well, I would say that the right hand side is trace(BA), but that is always equal to trace(AB).
     
  8. Sep 23, 2015 #7

    gill1109

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    Thank you! You are right.
     
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