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Math Help - Double Summations

  1. #1
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    Double Summations

    How would you expand the double summation:

    \sum\limits_{i}^2{\sum\limits_{j}^2{y_{i}y_{j}a_{i  j}}}

    I understand that:

    \sum\limits_{i}^2{\sum\limits_{j}^2{y_{i}y_{j}}}

    Gets you something like:

    <br />
y_{1}^{2}+y_{12}+ y_{21}+y_{2}^{2}<br />

    and

    \sum\limits_{i}^2{\sum\limits_{j}^2{a_{ij}}}

    is

    <br />
a_{11}+a_{12}+a_{21}+a_{22}<br />

    But how can you combine these two? Is it as simple as foiling the entire thing?
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  2. #2
    Member Haven's Avatar
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    If you're familiar with programming, double summations work like nested for loops. Ever time we iterate the outer summation, we run the entire inner summation. So every time i increases by 1, we go through all the possible values of j. In this case, every time i increases, we sum j from 1 to 2.

    So \sum_i^2 \sum_j^2 y_iy_ja_{ij} = \sum_j^2 y_1y_ja_{1j} + \sum_j^2 y_2y_ja_{2j} = (y_1)^2a_{11} + y_1y_2a_{12} + y_2y_1a_{21} + (y_2)^2a_{22}
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  3. #3
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    Also asked and answered here: http://www.mathhelpforum.com/math-he...ummations.html

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