# 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 **R**^{m}.

## 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).

## See also

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

## References

- James E. Gentle (2007).
*Matrix Algebra: Theory, Computations, and Applications in Statistics*. Springer-Verlag. p. 299. ISBN 0-387-70872-3.

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