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Linear Algebra Question (Kronecker Delta?)

  1. Sep 19, 2013 #1

    FeDeX_LaTeX

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    The problem statement, all variables and given/known data

    For fixed m ≥ 1, let ##\epsilon(i,j)## denote the m x m matrix ##\epsilon(i,j)_{rs} = \delta_{ir}\delta_{js}##, where i,j may denote any integers in the range 1 ≤ i,j ≤ m.

    (a) When m = 4, write out all ##\epsilon(i,j)## explicitly and label them correctly.

    The attempt at a solution

    My course hasn't started yet and all I know about the Kronecker delta/Levi-Civita epsilon notation is what little I've seen in videos.

    They haven't stated the parameters of r and s, but I guess that doesn't matter since ##\delta_{i,j} = 1## if and only if i = j, and 0 otherwise?

    All I can think of at the moment is this (can anyone tell me if this is correct?):

    ##\epsilon(1,1)_{rs} = \delta_{1r}\delta_{1s} = \delta_{11}\delta_{11}##
    ##\epsilon(1,2)_{rs} = \delta_{1r}\delta_{2s} = \delta_{11}\delta_{22}##
    ##\epsilon(1,3)_{rs} = \delta_{1r}\delta_{3s} = \delta_{11}\delta_{33}##
    (...) etc. making 16 values in total. Is this the kind of thing they're looking for?

    I still can't quite understand what I'm doing -- how does ##\delta_{11}\delta_{22}## represent an entry in a 4 x 4 matrix, for instance? Does that mean that both the entries in row 1 column 1 AND row 2 column 2 have value 1?

    (I apologise for deleting the 'relevant equations' section -- I literally don't know which ones are relevant.)
     
  2. jcsd
  3. Sep 19, 2013 #2

    tiny-tim

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    Hi FeDeX_LaTeX! :smile:
    No, they want you write out eg ##\epsilon(1,1)## as a matrix:

    ##\epsilon(1,1)_{11}\ \ \epsilon(1,1)_{12}##
    ##\epsilon(1,1)_{21}\ \ \epsilon(1,1)_{22}## :wink:
     
  4. Sep 19, 2013 #3

    FeDeX_LaTeX

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    Thanks for the reply -- I'm confused... how did you know ##\epsilon(1,1)## should be a 2 x 2 matrix?
     
  5. Sep 19, 2013 #4

    Ray Vickson

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    He does not say that ##\epsilon(1,1)## is a 2 x 2 matrix; he just shows you some of what you need to compute. The question actually say m = 4, so you need a 4 x 4 matrix.
     
  6. Sep 19, 2013 #5

    FeDeX_LaTeX

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    So I'm completing a 4 x 4 matrix consisting of only ##\epsilon(1,1)##?
     
  7. Sep 19, 2013 #6

    Office_Shredder

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    [tex]\epsilon(1,1)[/tex] IS a 4x4 matrix. The r,s entry of it is given by [itex] \delta_{1r} \delta_{s1} [/itex].

    In particular you will notice that many of the entries of this matrix are zero. Can you identify what values of r and s make [itex] \delta_{1r} \delta_{s1} [/itex] non-zero?
     
  8. Sep 19, 2013 #7

    Ray Vickson

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    It reads to me like it wants you to write out all 16 of the 4 x 4 matrices, although I don't see what the point of doing that could possibly be---unless, maybe, there is a pattern that it is important to exploit later in some context.
     
  9. Sep 19, 2013 #8

    FeDeX_LaTeX

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    Sorry, I wasn't aware it represented a matrix -- I thought it just represented some number permutations. So ##\epsilon(1,1)## is just going to give me a 4 x 4 matrix with zero for every entry apart from the top-left, right?

    r = s = 1 makes it non-zero?
     
  10. Sep 19, 2013 #9

    Office_Shredder

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    That's correct. The other ones can be solved similarly
     
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