# Euclidean distance matrix

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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 Rm.

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