I was wondering if there exists a certain type of matrix multiplication that just multiplies row $i$ by column $i$. For example let be
$$ A=\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix} $$
I want to represent the following matrix:
$$ B=\begin{bmatrix} \sum_{i=1}^3 a_{i1}^2 & \sum_{i=1}^3 a_{i2}^2 & \sum_{i=1}^3 a_{i3}^2 \\ \end{bmatrix} $$
I can define $B=\operatorname{diag}(A^TA)$, but I was wondering if there exists a type of product for representing $B$.
At the top of my head it doesn't seem like such operation would exist! By just for fun you could try to see the properties that such operation should have, maybe you get something interesting out of it!
The only "less known", I put it in quotation marks just because it's not totally true that isn't known but respect to classic matrix multiplication it is, is the Kroneker product defined as: given two matrices $A$ and $B$, their Kronecker product is as follows $$A\otimes B=\left(\begin{matrix}a_{11}B&a_{12}B&\cdots&a_{1n}B\\a_{21}B&a_{22}B&\cdots&a_{2n}B\\\vdots&\vdots&\ddots&\vdots\\a_{n1}B&a_{n2}B&\cdots&a_{nn}B\end{matrix}\right)$$