Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 29J / d

Since $X=U{V}^T$ and $X^{T}X=V\Lambda V^T$,ThereforePut $w_i=u_i/\sqrt{{\Lambda }_{ii}}$. By part (c), the $w_i$ are normalized sample principal components forming an orthonormal set, andThis is the orthogonal projection onto a finite-dimensional subspace spanned by the $w_i$, so ${\hat{Y}}_{\mathrm{MLE}}$ is the unique closest point in that subspace to $Y$.