= Euclidean distance matrix criterion
{c}
{title2=$D\text{ is Euclidean}\iff-\tfrac12HD^{(2)}H\succeq0$}
For symmetric nonnegative $D$ with zero diagonal, a centered Euclidean realization implies $D_{ij}^2=G_{ii}+G_{jj}-2G_{ij}$ and hence the displayed double-centering identity. Conversely, if the double-centered matrix is <positive semidefinite>, factor it as $XX^T$. Its reconstructed squared distances equal the original squared distances: their difference has zero double-centering, so is of the form $\mathbf1 a^T+a\mathbf1^T$; the zero diagonal forces $a=0$. The minimum exact embedding dimension is $\operatorname{rank}G$.
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