If I have a matrix:
$$
A =
\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
\end{bmatrix}
$$
and a vector:
$$
B =
\begin{bmatrix}
b_1 \\
b_2 \\
\end{bmatrix}
$$
What is the name of the operation, which produces a vector:
$$
\begin{bmatrix}
b_1 \cdot a_{11} + b_1 \cdot a_{12} + b_1 \cdot a_{13} \\
b_2 \cdot a_{21} + b_2 \cdot a_{22} + b_2 \cdot a_{23} \\
\end{bmatrix}
$$
Best Answer
Since your operation is invariant under row-permutations of matrix $A$ and only the sum matters then given $A$ we have $$ \tilde{A} = \begin{pmatrix} a_{11} + a_{12} + a_{13} \\ a_{21} + a_{22} + a_{23} \end{pmatrix} $$ and $b = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix}$. Now your operation becomes the Hadamard product of $\tilde{A}$ and $b$.