I am wondering if the following vector operation is known/studied, or alternatively if it can be simplified somehow into matrix multiplications:
$x$ is a vector and $A$ is the resulting matrix. The definition of $A$ is such that it is symmetric and the diagonal is $x$.
I am interested in this operation since I would later want to find optimal solutions to problems like $\min_x||A-D||^2_F$ (let's say this is the Frobenius norm) for some given data matrix $D$. In other words, I would like to find a set of weights $x$ such that if I perform this operation it will induce a matrix $A$ that is as "close" as possible to some target matrix $D$. BTW if it helps, I can think of $x$ values as probabilities so they are between 0 and 1, which means $A$ values will also be between 0 and 1.
I am looking for insights into this representation or the optimization problem.