Euclidean distance matrix

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In mathematics, a Euclidean distance matrix is an n×n matrix representing the spacing of a set of n points in Euclidean space. If A is a Euclidean distance matrix and the points are defined on m-dimensional space, then the elements of A are given by

$\begin{array}{rll} A & = & (a_{ij}); \\ a_{ij} & = & ||x_i - x_j||_2^2 \end{array}$

where ||.||₂ denotes the 2-norm on R^m.

[edit] Properties

Simply put, the element a_ij describes the square of the distance between the i ^th and j ^th points in the set. By the properties of the 2-norm (or indeed, Euclidean distance in general), the matrix A has the following properties.

All elements on the diagonal of A are zero (i.e. is it a hollow matrix).
The trace of A is zero (by the above property).
A is symmetric (i.e. a_ij = a_ji).
a_ij^1/2 is less than or equal to a_ik^1/2 + a_kj^1/2 (by the triangle inequality)
$a_{ij}\ge 0$