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.
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).
- Adjacency matrix
- Distance geometry
- Distance matrix
- Euclidean random matrix
- Classical multidimensional scaling, a visualization technique that approximates an arbitrary dissimilarity matrix by a Euclidean distance matrix