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Linear Operators and Matrices

  1. Oct 24, 2014 #1
    Suppose we have linear operators A' and B'. We define their sum C'=A'+B' such that
    C'|v>=(A'+B')|v>=A'|v>+B'|v>.

    Now we can represent A',B',C' by matrices A,B,C respectively. I have a question about proving that if C'=A'+B', C=A+B holds. The proof is

    Using the above with Einstein summation convention,
    C|v>=A|v>+B|v>
    and so component i on each side matches. Then
    Cijvj=Aijvj+Bijvj
    which holds for any |v>, so
    C=A+B
    as this is how we define matrix addition.

    However, why couldn't I have written
    Cijvj=Aikvk+Bilvl
    because I have changed only dummy variables, not affecting the sum. This would then not lead to Cijvj=Aijvj+Bijvj. I'm assuming it's something to do with the fact the next step sort of stops this sum from happening anyway, but I'm not sure.
     
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  3. Oct 24, 2014 #2

    Orodruin

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    Yes, you could have written that, but it would change nothing since the only difference is in what you call the dummy variables. You could just rename them back and write it back on your original form.
     
  4. Oct 24, 2014 #3
    It's just that if I do say
    Cijvj=Aikvk+Bilvl
    and then I write
    Cij=Aik+Bil
    It looks like a different story, and we can't see that the k and l were initially dummy variables and so we can't simply say it's ok to change them both to j.

    Ahh, actually, if we do that then we can't cancel out the v components anymore from each of the sums can we, which answers my question...
     
  5. Oct 24, 2014 #4

    Orodruin

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    k and l are dummy variables, you cannot just take them away from the v, you would end up with an expression that does not make sense (you cannot have different free indices in Cij). However, you can rename them both to j (they are dummy indices after all) and then factorise the vj out.
     
  6. Oct 26, 2014 #5

    Fredrik

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    Use the definition of matrix multiplication: ##(AB)_{ij}=A_{ik}B_{kj}##.
    \begin{align}
    &C_{ij}v_j=(Cv)_i\\
    &A_{ik}v_k+B_{il}v_l=(Av)_i+(Bv)_i=(Av+Bv)_i
    \end{align} Since the left-hand sides are equal, the right-hand sides are equal, and we have ##Cv=Av+Bv=(A+B)v##. If this holds for all v, then ##C=A+B##.
     
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