Regarding the dyadic product (specifically, the outer product or tensor product), notated as $\otimes$, can you provide comment on its counterpart? I'm not sure if it has a name, or if it has matrix notation. The operator I am thinking of represents that we divide every term of a vector by every other term of another vector, effectively acting like the division counterpart of the outer product. E.g., if $A =(a_1, a_2, a_3)$ and $B=(b_1, b_2, b_3)$, then $$C=A ? B = \begin{pmatrix}a_1/b_1 & a_1/b_2 & a_1/b_3 \\ a_2/b_1 & a_2/b_2 & a_2/b_3 \\ a_3/b_1 & a_3/b_2 & a_3/b_3 \end{pmatrix}$$
So, does this operator have a name, e.g., outer quotient or diadic quotient, and does it have a matrix notation? If the tensor (indicial) notation for the outer product is $$C_{ij} = A_i B_j$$ would the tensor notation for the operation I'm asking about be $$C_{ij} = A_i (1/B_j)?$$