Suppose we have linear operators A' and B'. We define their sum C'=A'+B' such that(adsbygoogle = window.adsbygoogle || []).push({});

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

C_{ij}v_{j}=A_{ij}v_{j}+B_{ij}v_{j}

which holds for any |v>, so

C=A+B

as this is how we define matrix addition.

However, why couldn't I have written

C_{ij}v_{j}=A_{ik}v_{k}+B_{il}v_{l}

because I have changed only dummy variables, not affecting the sum. This would then not lead to C_{ij}v_{j}=A_{ij}v_{j}+B_{ij}v_{j}. 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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# Linear Operators and Matrices

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