Let us refer to $D$ as the matrix with $D_{ij} = ||x_i - x_j||_2$ and $H = D \circ D$ e.g. the matrix with $H_{ij} = ||x_i - x_j||_2^2$ as its entries. Are the spectra of these matrices related at all? I suppose this question could be stated more generally in terms of Hadamard products.
In many papers, such as this one, $H$ is called a Euclidean Distance Matrices (EDM). Why? This matrix contains squared distances, not distances.
It seems that $H$ is more interesting to study for many reasons, such as the relationship between PCA and MDS. It is also clear that $H$ has been studied quite thoroughly.
This unaccepted answer is the only thing I can find which appears to relate these matrices, but, I am skeptical that it is correct. In particular, I believe when they write $D^2$ they mean $H$. If it is not a mistake, can someone clarify the definition of the matrix J? Unfortunately I cannot comment on the thread to ask this since I do not have enough reputation.
