In many papers, such as [this one][1] Euclidean Distance Matrices (EDM's) are referred to as it's Hadamard square. Why? This matrix is not the euclidean distance matrix.

Lets 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 spectrums of these matrices related at all? I suppose this question could be stated more generally in terms of hadamard products.

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][2] 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. 
 

  [1]: http://ac.els-cdn.com/002437959400031X/1-s2.0-002437959400031X-main.pdf?_tid=7a2c274a-0812-11e5-9a53-00000aab0f27&acdnat=1433131295_1b03b2056858f01d1059b3438672a703
  [2]: http://mathoverflow.net/questions/201027/how-can-we-interpret-the-eigenvalues-and-eigenvectors-of-euclidean-distance-matr