Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 9 / b / ii / Solution

Let $X_c$ be the column-centered data matrix. Its sample covariance matrix is $S=X_c^T{X}_c/(n-1)$. The loadings matrix $V=(v_1,\dots ,v_p)$ has orthonormal eigenvectors of $S$ as columns, ordered by nonincreasing eigenvalues:Each principal component loading specifies the coefficients of a principal direction in the original variables. Signs are arbitrary, and equal eigenvalues permit orthogonal changes of basis within their eigenspaces.