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 are defined on m-dimensional space, then the elements of A are given by where ||.||2 denotes the 2-norm on Rm.

Properties

Simply put, the element 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. ).
• (by the triangle inequality)
• • The number of unique (distinct) non-zero values within an n-by-n Euclidean distance matrix is bounded above by 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 are in general position, the rank is exactly min(n, m + 2).