# 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:

• $X_{ii}=0$;

• $X_{ij}=X_{ji}\geq 0$;

• $\sqrt{X_{ik}}\leq\sqrt{X_{ij}}+\sqrt{X_{jk}}$.

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 EuclideanDistanceMatrix 2013-03-22 14:37:15 2013-03-22 14:37:15 mathcam (2727) mathcam (2727) 7 mathcam (2727) Definition msc 15A48