1 Answers
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 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} in k-dimensional space ℝ, the elements of their Euclidean distance matrix A are given by squares of distances between them.That is
where ‖ ⋅ ‖ {\displaystyle \|\cdot \|} denotes the Euclidean norm on ℝ.
In the context of distance matrices, the entries are usually defined directly as distances, not their squares.However, in the Euclidean case, squares of distances are used to avoid computing square roots and to simplify relevant theorems and algorithms.
Euclidean distance matrices are closely related to Gram matrices.The latter are easily analyzed using methods of linear algebra.This allows to characterize Euclidean distance matrices and recover the points x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} that realize it.A realization, if it exists, is unique up to rigid transformations, i.e. distance-preserving transformations of Euclidean space.