- Euclidean distance matrix
In

mathematics , a**Euclidean distance matrix**is an "n×n" matrix representing the spacing of a set of "n" points inEuclidean 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:$egin\{array\}\{rll\}A\; =\; (a\_\{ij\});\backslash a\_\{ij\}\; =\; ||x\_i\; -\; x\_j||\_2^2end\{array\}$

where ||.||

_{2}denotes the2-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. is it a

hollow matrix ).

* The trace of "A" is zero (by the above property).

* "A" is symmetric (i.e. "a_{ij}" = "a_{ji}").

* "a_{ij}"^{1/2}is less than or equal to "a_{ik}"^{1/2}+ "a_{kj}"^{1/2}(by thetriangle inequality )

* $a\_\{ij\}ge\; 0$**References***; chapter 4.

*

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2010.*

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