# Euclidean distance matrix

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 $x_1,x_2,\ldots,x_n$ 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 Rm.

## Properties

Simply put, the element aij 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. it is a hollow matrix).
• The trace of A is zero (by the above property).
• A is symmetric (i.e. aij = aji).
• aij1/2 $\le$ aik1/2 + akj1/2 (by the triangle inequality)
• $a_{ij}\ge 0$
• The number of unique (distinct) non-zero values within an N-by-N Euclidean distance matrix is bounded (above) by [N*(N-1)] / 2 due to the matrix being symmetric and hollow.
• In dimension m, a Euclidean distance matrix has rank less than or equal to m+2. If the points $x_1,x_2,\ldots,x_n$ are in general position, the rank is exactly m+2.