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{aligned}A&=(a_{ij});\\a_{ij}&=d_{ij}^{2}\;=\;\lVert x_{i}-x_{j}\rVert _{2}^{2}\end{aligned}}

where ||.||₂ denotes the 2-norm on R^m.

A={\begin{bmatrix}0&d_{12}^{2}&d_{13}^{2}&\dots &d_{1n}^{2}\\d_{21}^{2}&0&d_{23}^{2}&\dots &d_{2n}^{2}\\d_{31}^{2}&d_{32}^{2}&0&\dots &d_{3n}^{2}\\\vdots &\vdots &\vdots &\ddots &\vdots &\\d_{n1}^{2}&d_{n2}^{2}&d_{n3}^{2}&\dots &0\\\end{bmatrix}}

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. $a_{{ij}}=a_{{ji}}$ ).
${\sqrt {a_{{ij}}}}\leq {\sqrt {a_{{ik}}}}+{\sqrt {a_{{kj}}}}$ (by the triangle inequality)
$a_{{ij}}\geq 0$
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