Solution to a matrix optimisation problem with a particular structure

Question

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.

Answer 1

I only have an answer for the first question: A matrix $A$ with entries $A_{ij} = v_i+w_j$ is called on outer sum of $v$ and $w$. I would write this as $$A = v\oplus w := v\otimes 1 + 1\otimes w := v1^T + 1w^T$$ where the $1$ denote "all-ones" vectors of appropriate size and $\otimes$ is the tensor product (also called outer product, hence the name outer sum).

In your case $v=w$ and $A$ depends linearly on $v$, so you can formulate your optimization problem directly in $v$ and take the derivatives w.r.t. $v$…

Answer 2

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.