On the uniqueness of Euclidean distance matrix completions: the case of points in general position
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.
scientific article; zbMATH DE number 2165362
scientific article