On yielding and jointly yielding entries of Euclidean distance matrices

Abstract

An $n \times n$ matrix D is a Euclidean distance matrix (EDM) if there exist $p^1, \ldots, p^n$ in some Euclidean space such that $d_{ij} = || p^i - p^j||^2$ for all $i,j=1,\ldots,n$. Let D be an EDM and let $E^{ij}$ be the $n \times n$ symmetric matrix with 1's in the $ij$th and $ji$th entries and 0's elsewhere. We say that $[l_{ij},u_{ij}]$ is the yielding interval of entry $d_{ij}$ if it holds that $D+t E^{ij}$ is an EDM iff $l_{ij} \leq t \leq u_{ij}$. If the yielding interval of entry $d_{ij}$ has length 0, i.e., if $l_{ij}=u_{ij}$, then $d_{ij}$ is said to be unyielding. Otherwise, if $l_{ij} \neq u_{ij}$, then $d_{ij}$ is said to be yielding. Let $d_{ij}$ and $d_{ik}$ be two unyielding entries of $D$. We say that $d_{ij}$ and $d_{ik}$ are jointly yielding if $D+t_1 E^{ij} + t_2 E^{ik}$ is an EDM for some nonzero scalars $t_1$ and $t_2$. In this paper, we characterize the yielding and the jointly yielding entries of an EDM D in terms of Gale transform of $p^1,\ldots,p^n$. Moreover, for each yielding entry, we present explicit formulae of its yielding interval. Finally, we specialize our results to the case where $p^1,\ldots,p^n$ are in general position.