How do you call call the operation that takes all possible sums of two elements from two vectors?

Question

Take two vectors $a\equiv (a_1,...a_n)$ and $b\equiv (b_1,...,b_m)$. Is there any name and symbol for denoting the vector that lists all possible sums of one element of $a$ and one element of $b$? That is, $$ \begin{pmatrix} a_1+b_1\\ a_2+b_1\\ \vdots\\ a_n+b_m\\ \vdots\\ a_1+b_m\\ a_2+b_m\\ \vdots\\ a_n+b_1\\ \end{pmatrix} $$

Answer 1

Following up on Exodd's comment. Define $E_k$ as the $k$-dimensional vector with all components equal to 1. Then take the sum of the outer products $(a\otimes E_m) + (E_n\otimes b)$ which is the $n \times m$ matrix $A_{ij} = a_i + b_j$. This is the transpose of Exodd's answer in the comments.

Update: As pointed out by Exodd, replace the Kronenecker product for the outer product in order to get a column vector.