The n body matrix and its determinant

Darij Grinberg, Peter J. Olver

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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 ≤ 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.

Original languageEnglish (US)
Pages (from-to)67-86
Number of pages20
JournalSIAM Journal on Applied Algebra and Geometry
Issue number1
StatePublished - 2019

Bibliographical note

Publisher Copyright:
© 2019 Society for Industrial and Applied Mathematics


  • Cayley–Menger matrix
  • Determinant
  • Distance geometry
  • Euclidean distance matrix
  • N body matrix
  • N body problem


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