Suppose we have two vectors $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^m$. I could define the mapping $$ T: \mathbb{R}^n\times \mathbb{R}^m \rightarrow \mathbb{R}^{n\times m} $$ as follows $$ T(x,y) = ( x_i+y_j )_{i,j=1}^{n,m}, $$ i.e. $T(x,y)$ is $n\times m$ matrix with elements equal sum of corresponding vectors. If instead of $x_i+y_j$ one considers $x_i \cdot y_j$ then it is matrix multiplication of two vectors ($x\cdot y^T$, where $x,y$ are column vectors). But what about sum?

Is that operation known and studied somewhere?