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Journal article
The n Body Matrix and Its Determinant
SIAM journal on applied algebra and geometry, v 3(1), 67
01 Jan 2019
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Abstract
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.
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Details
- Title
- The n Body Matrix and Its Determinant
- Creators
- Darij Grinberg - University of MinnesotaPeter J. Olver - University of Minnesota System
- Publication Details
- SIAM journal on applied algebra and geometry, v 3(1), 67
- Publisher
- Siam Publications
- Number of pages
- 20
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000462647900004
- Scopus ID
- 2-s2.0-85090625032
- Other Identifier
- 991021862239304721
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- Web of Science research areas
- Mathematics, Applied