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We can use $\sum_{i = 1}^{m}\sum_{j = 1}^{n} A_{ij}$ to denote the sum of all the elements in matrix $A_{m \times n}$. Now I would like to denote the sum of all the elements except $A_{rc}$. I come up with 2 options: $$ \sum_{\substack{i = 1 \\ i \neq r}}^{m} \sum_{\substack{j = 1 \\ j \neq c}}^{n} A_{ij} $$ and $$ \sum_{\substack{1 \leqslant i \leqslant m, 1 \leqslant j \leqslant n \\ i \neq r \land j \neq c}} A_{ij} $$ I don't know whether my second option looks weird. And I am wondering if my first choice denotes exactly what I want. Otherwise, are there any better choices?

Thanks in advance!

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If $\mathbf{e}$ is the column vector where all entries are 1's then you can represent the sum of all the entries of $A$ as $\mathbf{e}^TA\mathbf{e}$. To exclude an indice you can then subtract that indice as so: $$\mathbf{e}^TA\mathbf{e} - A_{rc}$$ Different forms will be more useful or intuitive than others depending on context.

crubow
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