Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 218 / 4 / c

The empirical kernel covariance operator $\Sigma$ is self-adjoint and positive semidefinite. Maximize $⟨u,\Sigma u⟩$ subject to $⟨u,u⟩=1$. The first variation of the Lagrange multiplier functional givesso $\Sigma v=\lambda v$. Taking the inner product with $v$ gives

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