Some Properties for the Euclidean Distance Matrix and Positive Semidefinite Matrix Completion Problems

Authors:
Hong-Xuan Huang;Zhi-An Liang;Panos M. Pardalos
Affiliations:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084,P.R. China/;Department of Mathematics, College of Sciences and Technology, Inner Mongolia University, Hohhot 010021, P.R. China/;Department of Industrial and Systems Engineering,University of Florida, Gainesville,FL 32611,USA
Venue:
Journal of Global Optimization
Year:
2003

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Abstract

The Euclidean distance matrix (EDM) completion problem and the positive semidefinite (PSD) matrix completion problem are considered in this paper. Approaches to determine the location of a point in a linear manifold are studied, which are based on a referential coordinate set and a distance vector whose components indicate the distances from the point to other points in the set. For a given referential coordinate set and a corresponding distance vector, sufficient and necessary conditions are presented for the existence of such a point that the distance vector can be realized. The location of the point (if it exists) given by the approaches in a linear manifold is independent of the coordinate system, and is only related to the referential coordinate set and the corresponding distance vector. An interesting phenomenon about the complexity of the EDM completion problem is described. Some properties about the uniqueness and the rigidity of the conformation for solutions to the EDM and PSD completion problems are presented.