The empirical <kernel covariance operator> $\Sigma$ is self-adjoint and positive semidefinite. Maximize $\langle u,\Sigma u\rangle$ subject to $\langle u,u\rangle=1$. The first variation of the <Lagrange multiplier> functional gives
$$
2\Sigma v-2\lambda v=0,
$$
so $\Sigma v=\lambda v$. Taking the inner product with $v$ gives
$$
\lambda=\langle v,\Sigma v\rangle
=\frac1n\sum_{i=1}^n|\langle v,\phi(x_i)\rangle|^2\geq0.
$$

Solved by gpt-5.6-sol high.

Solution

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

{title2=$\Sigma$}

For feature vectors $\phi(x_i)$, the empirical uncentered kernel covariance operator is
$$
\Sigma v=\frac1n\sum_i\phi(x_i)\langle\phi(x_i),v\rangle.
$$


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