Question:
is there a name for the following operation
$$\sum_{i=1}^n\sum_{j=1}^mx_iy_j^T,\ x_i,y_j\in \mathbb{R}^k$$ i.e. for generating a square matrix that is the sum of the cartesian product of a set $X$ of column vectors with a set $Y$ of row vectors?
The reason for asking is that it came up in the solution of a problem.
The name for the above operation is 2nd rank tensor composition / matrix multiplication.
I will explain in a matrix language, since it is easier to grasp. Assume there is an orthonormal basis in which all vectors $ \mathbf{x}_{k},\mathbf{y}_{k} $ are expressed
If you organize the $ \mathbf{x}_{k}$ vectors as columns of an $n\times n$ matrix, say $\mathbf{A}$, and the $ \mathbf{y}_{k}$ vectors as the rows of another $n\times n$ matrix, say $\mathbf{B}$, then the expression above is exactly the product $\mathbf{AB}$.
This is based on the fact that the product of any two matrices $\mathbf{A,B}$ can be seen as: $$\mathbf{AB} = \sum_{k=1}^{N} \mathbf{c}_k^{\mathbf{A}} \otimes \mathbf{l}_k^{\mathbf{B}},$$ where $\mathbf{c}_k^{\mathbf{A}}$ are the columns of $\mathbf{A}$ and $\mathbf{l}_k^{\mathbf{B}}$ are the rows of $\mathbf{B}$, while $N$ denotes the number of columns of $A$ (the same as the number of rows of $B$, such that the multiplication could be performed)
