Does a matrix of the form $A\_{ij} = v\_i + v\_j$ for some arbitrary vector $v$ have a particular name?


I am attempting to find the closed form solution (if it exists, although it *looks* like it might) for the $v$ that solves the optimisation problem


$ \text{min}\_v ||A - M||\_F$


for an arbitrary matrix $M$, where $||.||\_F$ is the Frobenius norm. The issue I am having is that, in taking the derivative with respect to $v\_i$ and setting it to zero, it becomes apparent that $A\_{ij} = v\_i + v\_j$ is an abuse of notation.



Solution to a matrix optimisation problem with a particular structure
Solved:


$v\_i = \frac{1}{N} \left( m\_i - \frac{1}{2N} \sum\_j m\_j \right)$


where $m\_i = \frac{1}{2} \sum\_j M\_{ij} + M\_{ji}$ and $N$ is the dimension of the space.


