Some necessary and some sufficient trace inequalities for Euclidean distance matrices

Abstract

In this article, we use known bounds on the smallest eigenvalue of a symmetric matrix and Schoenberg's theorem to provide both necessary as well as sufficient trace inequalities that guarantee a matrix D is a Euclidean distance matrix, EDM . We also provide necessary and sufficient trace inequalities that guarantee a matrix D is an EDM generated by a regular figure.

Keywords:

• Euclidean distance matrices
• Smallest eigenvalue
• AMS Subject Classifications
• 15A45
• 15A57

Acknowledgements

Research supported by the Natural Sciences and Engineering Research Council of Canada and MITACS.

Notes

¹ Some authors refer to these as EDMs of strength one 6.