Euclidean distance matrix

A Euclidean distance matrix (EDM) is a real $m\times m$ matrix $X$ such that for some points $y_{1},\ldots,y_{m}$ in $\mathbb{R}^{m}$ , $X_{ik}=\left\|y_{i}-y_{k}\right\|_{2}^{2}$ , where $\|\cdot\|_{2}$ is the 2-norm 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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$X_{ij}=X_{ji}\geq 0$ ;
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$\sqrt{X_{ik}}\leq\sqrt{X_{ij}}+\sqrt{X_{jk}}$ .

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, $z^{T}Xz\leq 0$ .

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	2013-03-22 14:37:15
Last modified on	2013-03-22 14:37:15
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	7
Author	mathcam (2727)
Entry type	Definition
Classification	msc 15A48