Euclidean distance matrix
A Euclidean distance matrix (EDM) is a real $m\times m$ matrix $X$ such that for some points ${y}_{1},\mathrm{\dots},{y}_{m}$ in ${\mathbb{R}}^{m}$, ${X}_{ik}={\parallel {y}_{i}{y}_{k}\parallel}_{2}^{2}$, where $\parallel \cdot {\parallel}_{2}$ is the 2norm on ${\mathbb{R}}^{m}$.
A EDM $X$ inherits the following from the norm that defines it:

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${X}_{ii}=0$;

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${X}_{ij}={X}_{ji}\ge 0$;

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$\sqrt{{X}_{ik}}\le \sqrt{{X}_{ij}}+\sqrt{{X}_{jk}}$.
Additionally, $X$ is a EDM if and only if the diagonal entries of $X$ are all 0 and for all $z\in {\mathbb{R}}^{m}$ whose components^{} sum to 0, ${z}^{T}Xz\le 0$.
Finally, the set of $m\times m$ EDMs forms a convex cone (http://planetmath.org/Cone3) in the set of all $m\times m$ matrices.
References
 1 S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
Title  Euclidean distance matrix 

Canonical name  EuclideanDistanceMatrix 
Date of creation  20130322 14:37:15 
Last modified on  20130322 14:37:15 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  7 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 15A48 