Sorry if the question is lame, but I'm struggling to find the answer to the following problem: Given a matrix $A\in \mathbb{R}^{n,n}$ and a column vector $b\in \mathbb{R}^{n}$, how can one find the values of vectors $u \in \mathbb{R}^{n}$ such that $$ b = A\circ u u^T \mathbb{1}$$
where $\circ$ is the elementwise (Hadamard) product, $\mathbb{1}$ is a column vectors with ones and $x y^T$ is the outer product between $x$ and $y$. This means that for the $i_{th}$ element we can write: $$b_i = \sum_{j=1}^n a_{i,j} u_i u_j $$ The problems is quadratic, but the number of variables is equal to $n$, so I think there should be an easy solution to this class of problems.
Mybe the problem is known with some particular name, but I cannot find it in the literature. Hope you have some suggestions, thanks!