Euclidean distance matrix

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

:egin{array}{rll}A & = & (a_{ij});\a_{ij} & = & ||x_i - x_j||_2^2end{array}

where ||.||2 denotes the 2-norm on Rm.


Simply put, the element "aij" 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. is it a hollow matrix).
* The trace of "A" is zero (by the above property).
* "A" is symmetric (i.e. "aij" = "aji").
* "aij"1/2 is less than or equal to "aik"1/2 + "akj"1/2 (by the triangle inequality)
* a_{ij}ge 0


*; chapter 4.

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