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| In [[mathematics]], a '''Euclidean distance matrix''' is an ''n×n'' [[matrix (mathematics)|matrix]] representing the spacing of a set of ''n'' [[point (geometry)|points]] in [[Euclidean space]]. If ''A'' is a Euclidean distance matrix and the points <math>x_1,x_2,\ldots,x_n</math> are defined on ''m''-dimensional space, then the elements of ''A'' are given by
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| :<math>\begin{array}{rll}
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| A & = & (a_{ij});
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| \\
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| a_{ij} & = & ||x_i - x_j||_2^2
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| \end{array}
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| </math>
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| where ||.||<sub>2</sub> denotes the [[2-norm]] on '''R'''<sup>m</sup>.
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| ==Properties==
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| Simply put, the element ''a<sub>ij</sub>'' describes the square of the distance between the ''i''<sup> th</sup> and ''j''<sup> th</sup> points in the set. By the properties of the 2-norm (or indeed, Euclidean distance in general), the matrix ''A'' has the following properties.
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| * All elements on the [[diagonal of a matrix|diagonal]] of ''A'' are zero (i.e. it is a [[hollow matrix]]).
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| * The [[trace of a matrix|trace]] of ''A'' is zero (by the above property).
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| * ''A'' is [[symmetric matrix|symmetric]] (i.e. ''a<sub>ij</sub>'' = ''a<sub>ji</sub>'').
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| * ''a<sub>ij</sub>''<sup>1/2</sup> <math>\le</math> ''a<sub>ik</sub>''<sup>1/2</sup> + ''a<sub>kj</sub>''<sup>1/2</sup> (by the [[triangle inequality]])
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| * <math> a_{ij}\ge 0</math>
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| * 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.
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| * In dimension ''m'', a Euclidean distance matrix has [[Rank (linear algebra)|rank]] less than or equal to ''m+2''. If the points <math>x_1,x_2,\ldots,x_n</math> are in [[General_position| general position]], the rank is exactly ''m+2''.
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| ==See also==
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| * [[Adjacency matrix]]
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| * [[Distance matrix]]
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| * [[Euclidean random matrix]]
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| ==References==
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| * {{cite book | author=James E. Gentle | title=Matrix Algebra: Theory, Computations, and Applications in Statistics | publisher=[[Springer-Verlag]] | date=2007 | isbn=0-387-70872-3 | page=299 }}
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| [[Category:Matrices]]
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| {{geometry-stub}}
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