Euclidean Distance Matrices
Imagine you have 
points in 
, then construct the ‘Euclidean distance matrix’ 
. 
has some interesting properties: all its entries are positive, it is hollow (the diagonal entries are all zero), its entries satisfy 
, and most interesting, 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.
Possibly relevant posts:
- Interpolating splines (1/22/2006)
- Expected norm of matrices with randomly signed entries (8/6/2008)
- Boundedness of products of certain matrices (10/29/2007)
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.
Comment by andrew corrigan — 10/26/2007 @ 7:53 pm