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 R^m.

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. it is a hollow matrix).
The trace of A is zero (by the above property).
A is symmetric (i.e. $a_{ij} = a_{ji}$ ).
$\sqrt{a_{ij}} \le \sqrt{a_{ik}} + \sqrt{a_{kj}}$ (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 min(n, m + 2).

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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Euclidean distance matrix

Properties

See also

References