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Double summation

  1. Aug 18, 2012 #1
    If I know that [itex]\sum_{k=1}^n a_{ik} = 1[/itex] and [itex]\sum_{j=1}^n b_{kj} = 1[/itex], why is the following permitted?

    [tex]\sum_{j=1}^n \sum_{k=1}^n a_{ik}b_{kj} = \left(\sum_{j=1}^n b_{kj}\right) \left(\sum_{k=1}^n a_{ik}\right) = 1\cdot 1 = 1[/tex]


    Thanks!
     
  2. jcsd
  3. Aug 18, 2012 #2

    micromass

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    It's not permitted. What you wrote makes no sense since you sum over k and one of the [itex]b_{kj}[/itex] is outside that sum. That is not allowed.

    What you could do is:

    [tex]\sum_{j=1}^n\sum_{k=1}^n a_{ik}b{kj}= \sum_{k=1}^n \sum_{j=1}^n a_{ik}b_{kj} = \sum_{k=1}^n \left(a_{ik} \sum_{j=1}^n b_{kj}\right)= \sum_{k=1}^n a_{ik}=1[/tex]
     
  4. Aug 18, 2012 #3
    Thank you very much. I knew I wasn't understanding something.
     
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