Classical multidimensional scaling ($G=-\frac{1}{2}H{D}^{(2)}H,X_k=U_k{\Lambda }_k^{1/2}$)

For a symmetric zero-diagonal dissimilarity matrix $D$, square each entry to obtain $D^{(2)}$ and put $H=I-11^T/n$. Take the eigenvectors of $G=-H{D}^{(2)}H/2$ corresponding to its largest positive eigenvalues. Multiplying them by square roots of these eigenvalues gives coordinates. If $D$ contains Euclidean distances, $G$ is a centered Gram matrix, so retaining all positive eigenvalues reconstructs those distances exactly. Retaining fewer approximates the Gram matrix, not generally the sum of squared errors in raw distances. Negative eigenvalues signal failure of an exact Euclidean representation. Coordinates are unchanged in meaning by orthogonal transformations.

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 $X{X}^T$. Its reconstructed squared distances equal the original squared distances: their difference has zero double-centering, so is of the form $1a^T+a{1}^T$; the zero diagonal forces $a=0$. The minimum exact embedding dimension is $\mathrm{rank}G$.