Matrix from scaled vectors notation

Question

I'm trying to find a concise way to notate the following operation:

$$ \vec{a}^T \space \text{(Operator)} \space \vec{b} = \begin{bmatrix} a_1 \vec{b} \space \dots \space a_N \vec{b} \end{bmatrix} $$

for my personal work I have been using the Hadamard product ($\circ$) for (Operator), but I don't think that is correct. Is there a standard notation to write this?

Answer 1

The operation is tensor product $\otimes$ (or it's sometimes called outer product): $$ \vec b\otimes\vec a = \begin{pmatrix} a_1b_1&\cdots &a_nb_1\\ \vdots & \ddots & \vdots \\ a_1 b_n& \ldots& a_nb_n \end{pmatrix} $$