Concept

Euclidean distance matrix

Summary
In mathematics, a Euclidean distance matrix is an n×n matrix representing the spacing of a set of ''n'' points in Euclidean space. For points x_1,x_2,\ldots,x_n in ''k''-dimensional space ℝk, the elements of their Euclidean distance matrix ''A'' are given by squares of distances between them. That is :\begin{align} A & = (a_{ij}); \ a_{ij} & = d_{ij}^2 ;=; \lVert x_i - x_j\rVert^2 \end{align} where |\cdot| denotes the Euclidean norm on ℝk. :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} In the context of (not necessarily Euclidean) distance matrices, the entries are usually defined directly as distances, not their squares. However, in the Euclidean case, squares of distances
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