# Expressing the addition of vertex potentials via linear algebra

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.

Let $$\mathbf{d}$$ be the diagonal matrix with $$1,2,\ldots,n$$ on the diagonal. Denote by $$\pmb{\pi} \mapsto A\cdot \pmb{\pi}$$ the application of a linear map $$A\colon \mathbb{R}^{n\times n} \to \mathbb{R}^{n\times n}$$ on the space of matrices. In particular, denote $$[\mathbf{d},-]\cdot \pmb{\pi} = \mathbf{d}\pmb{\pi} - \pmb{\pi}\mathbf{d}$$. Then $$\operatorname{diag}(\pmb{\pi}) = \frac{\sin(2\pi[\mathbf{d},-])}{2\pi [\mathbf{d},-]} \cdot \pmb{\pi} ,$$ which is easy to check by diagonalizing $$[\mathbf{d},-]$$ over the space of matrices.