Inverse eigenvalue problem for Euclidean distance matrices

Abstract

This paper examines the inverse eigenvalue problem (IEP) for the particular class of Euclidean distance matrices. Studying the necessary and sufficient conditions for a matrix to be an element of the set of Euclidean distance matrices gives us a better understanding as to the necessary conditions placed on the numbers to be the eigenvalues of some Euclidean distance matrix. Using these necessary conditions, Hayden was able to solve the IEP order n using a Hadamard matrix of order n. After 10 years, an error in Hayden's construction of the n+1 solution to the IEP for Euclidean distance matrices was noted and corrected accordingly. However the result was not a solution to the IEP of order n+1.