Description Usage Arguments Details Value Examples

View source: R/MatrixOperators.R

`edm2gram`

Linear transformation of a Euclidean Distance Matrix to a Gram Matrix

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`D` |
A Euclidean Distance Matrix |

While we specify that the input should be a Euclidean Distance Matrix (as this results in a Gram Matrix) the domain of edm2gram is the set of all real symmetric matrices. This function is particularly useful as it has the following property:

*edm2gram(D_{n}^{-}) = B_{n}^{+}*

where *D_{n}^{-}* is the space of symmetric, hollow matrices, negative definite on the space spanned by *x'e = 0*
and *B_{n}^{+}* is the space of centered positive definite matrices.

We can combine these two properties with a well known result: If D is a real symmetric matrix with 0 diagonal (call this matrix pre-EDM),
then D is a Euclidean Distance Matrix iff D is negative semi-definite on *D_{n}^{-}*.

Using this result, combined with the properties of edm2gram we therefore have that
D is an EDM iff D is pre-EDM and *edm2gram{D}* is positive semi-definite.

G A Gram Matrix, where G = XX', and X is an nxp matrix containing the point configuration.

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