Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2024 / Paper 1 / E8 / e

The premise contains a wording error: $\lambda$ must be a nonzero eigenvalue, rather than a nonzero eigenvector. Let $v\ne 0$ satisfyThen $Xv\ne 0$, since otherwise $X^{T}Xv=0$ would force $\lambda v=0$. Multiplying by $X$ givesThus $\boxed{\lambda \text{ is an eigenvalue of }X{X}^T}$, with eigenvector $Xv$. Conversely, applying $X^T$ maps an eigenvector of $X{X}^T$ with nonzero eigenvalue to one of $X^{T}X$.

These maps are inverses up to the factor $\lambda$ on the corresponding eigenspaces, so the nonzero eigenvalues have equal multiplicities. Their zero eigenvalues need not have the same multiplicity because the two matrices can have different sizes. For a unit eigenvector $v$ and real $X$, the normalized partner is $u=Xv/\sqrt{\lambda }$, since $\Vert Xv{\Vert }^2=v^T{X}^{T}Xv=\lambda >0$.

For centred data, classical multidimensional scaling from Euclidean distances constructs the observation Gram matrixwhere $D_{ij}^{(2)}=\Vert x_i-x_j{\Vert }^2$. By part (i), its nonzero eigenvalues agree with those of $X^{T}X$, the matrix used for principal component analysis before dividing by a covariance normalization.

If $u_i=X{v}_i/\sqrt{{\lambda }_i}$ is the unit observation-space eigenvector, the corresponding classical multidimensional scaling coordinate column isThat is exactly the column of principal component scores. Consequently 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.