The $n$ Body Matrix and Its Determinant

The primary purpose of this note is to prove two recent conjectures concerning the $n$ body matrix that arose in recent papers of Escobar-Ruiz, Miller, and Turbiner on the classical and quantum $n$ body problem in $d$-dimensional space. First, whenever the positions of the masses are in a nonsingular configuration, meaning that they do not lie on an affine subspace of dimension $\leq n-2$, the $n$ body matrix is positive definite and, hence, defines a Riemannian metric on the space coordinatized by their interpoint distances. Second, its determinant can be factored into the product of the order $n$ Cayley--Menger determinant and a mass-dependent factor that is also of one sign on all nonsingular mass configurations. The factorization of the $n$ body determinant is shown to be a special case of an intriguing general result proving the factorization of determinants of a certain form.