Euclidean Distance Matrices

October 26th, 2007 ~ Posted in: Mathematics

Imagine you have m points in \R^n, then construct the ‘Euclidean distance matrix’ A_{ij} = \|p_i - p_j\|. A has some interesting properties: all its entries are positive, it is hollow (the diagonal entries are all zero), its entries satisfy A_{ij} \leq A_{ik} + A_{kj} , and most interesting, A is nonsingular. The latter is particularly interesting– why it is true is an interesting question in its own right– because it makes EDMs a useful tool for interpolation.

Mathematics ~

One Response to “Euclidean Distance Matrices”

  • 1. andrew corrigan
    October 26th, 2007 at 7:53 pm

    Where does the nonsingularity come from? I’m pretty sure that the norm is just conditionally negative definite, so it doesn’t come from positive or negative definiteness.

    I prefer the Wendland functions, since they lead to sparse and positive definite interpolation matrices.

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