I have read here that:
$$ M \circ \vec{v} = \operatorname{Diag}(\vec{v}) \, M $$
That is, the Hadamard product between an $n\times m$ matrix and an $n \times 1$ vector is equivalent to the dot product between a diagonal matrix whose main diagonal is the vector and the $n\times m$ matrix.
However, I cannot find this definition anywhere else. What is the proof and source of this?
Note that the Hadamard product is not typically defined for matrix of different size. In this context, however, it looks like multiplication is being "broadcast" over the columns of the matrix. That is, for a matrix $M$ with columns $\vec M_1,\dots,\vec M_n$, we have $$ M \circ \vec v = [\vec M_1 \ \ \cdots\ \ \vec M_m] \circ \vec v = [\vec M_1 \circ \vec v \ \ \cdots\ \ \vec M_m \circ \vec v]. $$ This can be rewritten as a more standard Hadamard product as follows. Let $\vec e = (1,1,\dots,1) \in \Bbb R^n$. Then $\vec v \vec e^T$ is a matrix with the same shape as $M$, and $$ M \circ (\vec v \vec e^T) = M \circ \vec v. $$ With that, you can now use the following standard result: $$ M \circ (\vec v \vec e^T) = \operatorname{Diag}(\vec v) M\operatorname{Diag}(\vec e), $$ and note that $\operatorname{Diag}(\vec e)$ is the identity matrix so that we indeed have $M \circ (\vec v \vec e^T) = \operatorname{Diag}(\vec v) M$ as expected.