Circum-Euclidean distance matrices and faces
type: Item
instance of: scholarly article
title: Circum-Euclidean distance matrices and faces
zbMATH Open document ID: 0837.15014
DOI: 10.1016/0024-3795(94)00031-X
author: Pablo Tarazaga
author: Thomas L. Hayden
author: James H. Wells
published in: Linear Algebra and its Applications
publication date: 1996-05-13T00:00:00Z
review text: A nonnegative symmetric \(n \times n\) matrix \(D = (d_{ij}^2)\) with zeros on the diagonal is called a dissimilarity matrix. If there exist \(n\) points \(p_1, \dots, p_n\) in \(\mathbb{R}^r\) such that \(d_{ij}^2 = |p_i - p_j |^2\), then \(D\) is called Euclidean distance matrix where \(|p - q |\) denotes the usual Euclidean distance. The smallest such value of \(r\) is called the embedding dimension. \(D\) is called a circum-Euclidean provided the points which generate it lie on the surface of some hypersphere. The authors study the structure of circum-Euclidean distance matrices, they show that such Euclidean distance matrices are characterized as having constant row sums and they constitute the interior of the cone of all Euclidean distance matrices. The authors provide a formula for computing the radius of a representing configuration in the smallest embedding dimension \(r\) and show that \(rk D = r + 1\). Finally, they present a geometric characterization of the faces of this cone. Given a configuration of points and its Euclidean distance matrix \(D\), any matrix in the minimal face containing \(D\) comes from a configuration that is a linear perturbation of the points that generate \(D\).
reviewed by: K.Burian
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 841755
zbMATH Keywords: cone of Euclidean distance matrices
zbMATH Keywords: dissimilarity matrix
zbMATH Keywords: circum- Euclidean distance matrices
zbMATH Keywords: smallest embedding dimension
zbMATH Keywords: faces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 841755