= Solution
For centred data, <classical multidimensional scaling> from <Euclidean distances> constructs the observation <Gram matrix>
$$
B=-\tfrac12HD^{(2)}H=XX^T,\qquad H=I-\tfrac1n\mathbf1\mathbf1^T,
$$
where $D^{(2)}_{ij}=\|x_i-x_j\|^2$. By part (i), its nonzero <eigenvalues> agree with those of $X^TX$, the <matrix> used for <principal component analysis> before dividing by a covariance normalization.
If $u_i=Xv_i/\sqrt{\lambda_i}$ is the unit observation-space <eigenvector>, the corresponding <classical multidimensional scaling> coordinate column is
$$
\boxed{\sqrt{\lambda_i}\,u_i=Xv_i}.
$$
That is exactly the column of <principal component scores>. Consequently \b[<classical multidimensional scaling> of Euclidean pairwise distances recovers the same reduced coordinates as <principal component analysis>], up to signs and arbitrary <orthogonal transformations> within equal-<eigenvalue> subspaces. Distances alone cannot fix translation or orientation. The equivalence presumes centred data and Euclidean distances; general dissimilarities need not yield a positive semidefinite <Gram matrix> or a corresponding data <covariance matrix>.
Back to article page
