Re: mulling metrics

Let's have four points a,b,c,d having distances d(ij) = 1. They can be
arranged into a regular tetrahedron.
If we arrange them as a square, than two distances must be square roots
of 2, if as a cube, than three distances must be square roots of 2 or
one distance must be square roots 3.
If we omit square roots, we use squared distances.
Now we write distance matrices with elements d(ij). How we express the
fact that four points a,b,c,d lie on a straight line? We must use
squared distances, e.g. d(ad) = 9.
Than distance matrices straight lines have only two nonzero
eigenvalues, and similarly, the number of nonzero eigenvalues of
more dimensional objects is determined by the embedding dimension.
kunzmilan

.

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