Define a matrix operation

Question

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

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

Answer 1

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