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.

Properties

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

References

*; chapter 4.
*


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