Question:Given a $\pmb{A}\in\mathbb{R}^{n\times n},\ \pmb{A}^T=\pmb{A},\ \pmb{A}_{ii}=0$, is it possible, to generate via operations of linear algebra $\pmb{B}\in\mathbb{R}^{n\times n}:\ \pmb{B}_{ij}=\pmb{A}_{ij}+\pmb{\pi}_i+\pmb{\pi}_j,\ \pmb{B}_{ii}=0$, i.e. to formulate the adding of vertex potentials that are utilized in Held and Karp's calculation of high lower bounds on the length of the optimal solution of a symmetric traveling salesman problem?

It is fairly easy without the restriction of using operations from linear algebra and agreeing on column vectors:

$\pmb{\pi}\in\mathbb{R}^n,\ \pmb{e}\in\mathbb{R}^n,\pmb{e}_i=1\ \implies\ \pmb{B}=\pmb{A}+\pmb{\pi}\pmb{e}^T+\pmb{e}\pmb{\pi}^T-2 diag(\pmb{\pi})$, so the question boils down to as to whether it is possible to generate a diagonal matrix, whose diagonal elements resemble that of a given vector, from a vector with linear algebraic operations.

The existence of such an operation would be helpful in a proof of properties of vertex potentials that are calculated from the entries of $\pmb{A}$ or, put differently, whether it is possible to calculate a "ground state" of symmetric matrices by annihilating vertex potentials.

The following two images depict the effect of removing vertex potentials on the Minimum Spanning Tree of a planar pointset:

The red dotted lines depict the convex hull of the 3000 points; removing the vertex weights incurs some kind of shape sensitivity to the MST and it is remarkable that it contains a very long path that almost resembles a hull of the pointset.