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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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