Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 218 / 6 / c / i / Solution

For $\varphi (x)=x$ and centered data, the objective is the Rayleigh quotientThis is the first-direction optimization in principal component analysis. Therefore $\hat{u}$ is any unit eigenvector of the sample covariance matrix $S$ corresponding to its largest eigenvalue.