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Summation Convention for 2 Vectors

  1. Apr 5, 2014 #1
    From an exercise set on the summation convention: X and Y are given as [Xi] = \begin{pmatrix}
    1\\ 0\\ 0\\ 1\end{pmatrix} and [Yi] = \begin{pmatrix} 0\\ 1\\ 1\\ 1\end{pmatrix} There are a few questions involving these vectors. The one I am stuck on asks to compute XiYj .


    It may be necessary to raise lower indices in the question, the book that this question comes from uses a metric with signature ( - + + + ) for doing this.



    I have no attempt, I have no idea what the question actually wants. I thought there is only a summation if the indices are the same and matrix multiplication is obviously not an option
     
    Last edited by a moderator: Apr 5, 2014
  2. jcsd
  3. Apr 5, 2014 #2

    pasmith

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    [itex]x^iy^j[/itex] is the tensor of rank 2 (square matrix) which has [itex]x^i y^j[/itex] at row [itex]i[/itex], column [itex]j[/itex].
     
    Last edited by a moderator: Apr 5, 2014
  4. Apr 5, 2014 #3
    So in other words it's a YXT operation that's required. I am quite confused by your answer, is it supposed to be evident from the notation that this is the thing to do?
     
  5. Apr 5, 2014 #4

    Mark44

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    BHL 20, please don't adorn your text with size tags.
     
  6. Apr 6, 2014 #5

    Fredrik

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    Maybe you should post the exact problem statement. "Compute ##X^i Y^j##" doesn't make much sense to me. Does it mean "compute ##X^iY^j## for all i,j? Then the straightforward way is to just do these multiplications one at a time: ##X^0Y^0=1\cdot 0=0##, ##X^1Y^0=0\cdot 0=0##,... The results of these 16 calculations can be arranged in a matrix, which for all i,j, has ##X^iY^j## on row i, column j (and the results can be arranged in other ways as well), but you shouldn't be required to present the result in the form of a matrix.

    If you really want to, you can use the definition of matrix multiplication in the following way. For all i,j, we have
    $$(XY^T)^i{}_j =\sum_{k=0}^0 X^i{}_k (Y^T)^k{}_j = X^i{}_0 Y^j{}_0 =X^i Y^j.$$
     
  7. Apr 7, 2014 #6
    Thanks Fredrik, I see now.
     
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