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The matrix product is defined as $$(AB)_{ij}=\sum_{m}a_{im}b_{mj}$$

What kind of the opeation is what is below? $$(A ? B)_{ij}=\sum_{m,n}a_{im}b_{nj}$$

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The questions is somehow vague, so what follows is not a canonical answer.

$(A ? B)_{ij}=\sum_{m,n}a_{im}b_{nj}=\sum_{m}a_{im}\sum_nb_{nj}$

This gives the following interpretation $(A ? B)_{ij}$ is the product of two things: the sum of the $i$-th row of $A$ and the sum of the $j$-th column of $B$.

This may be rewritten as $$(A ? B)_{ij}=\left(A(1,1,1,\cdots, 1)^T\right)_{i1}\cdot\left((1,1,1,\cdots, 1)A\right)_{1j}$$

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  • $\begingroup$ Let ${\bf 1}$ be a rectangular matrix of ones, then the operation can be expressed as the matrix product $A\!\cdot{\bf 1}\!\cdot\!B$. The result is a rank-1 matrix. $\endgroup$
    – lynn
    Feb 28, 2015 at 0:37

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