On the uniqueness of Euclidean distance matrix completions: the case of points in general position
type: Item
instance of: scholarly article
title: On the uniqueness of Euclidean distance matrix completions: the case of points in general position
zbMATH Open document ID: 1072.15019
author: Abdo Y. Alfakih
published in: Linear Algebra and its Applications
publication date: 2005-05-04T00:00:00Z
review text: The paper deals with the Euclidean distance matrix completion problem. An \(n \times n\) matrix \(D=(d_{ij})\) is called an Euclidean distance matrix if there exist points \(p^1,p^2,\ldots,p^n\) in some Euclidean space such that \(d_{ij}=\| p^{i}-p^{j}\| ^2\) for all \(i,j=1,2,\ldots,n\). A matrix \(A=(a_{ij})\) is symmetric partial if some of its entries are specified, and \(a_{ji}\) is specified and equals to \(a_{ij}\) whenever \(a_{ij}\) is specified. Given an \(n \times n\) symmetric partial matrix \(A\), an \(n \times n\) matrix \(D\) is said to be a Euclidean distance matrix completion of \(A\) if \(D\) is an Euclidean distance matrix and \(d_{ij}=a_{ij}\) for all specified entries of \(A\). The Euclidean distance matrix completion problem asks which symmetric partial matrices have an Euclidean distance matrix completion. In this paper, the author analyzes necessary and sufficient conditions for the uniqueness of a given Euclidean distance matrix completion in the case where this completion is generated by points in general position. In addition, he shows that the problem of checking the validity of these conditions can be formulated as a semidefinite programming problem.
Mathematics Subject Classification ID: 15A29
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 51K05
Mathematics Subject Classification ID: 52B35
Mathematics Subject Classification ID: 90C22
zbMATH DE Number: 2165362
zbMATH Keywords: Euclidean distance matrix
zbMATH Keywords: Gale transform
zbMATH Keywords: semidefinite programming
zbMATH Keywords: matrix completion problems
zbMATH Keywords: points in general position
zbMATH Keywords: symmetric partial matrix
reviewed by: Juan Ramón Torregrosa Sánchez
describes a project that uses: SeDuMi
MaRDI profile type: Publication
full work available at URL: j.laa.2004.10.025
OpenAlex ID: W2087832775
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DOI: 10.1016/J.LAA.2004.10.025
description: scientific article; zbMATH DE number 2165362