A <positive-semidefinite kernel> on a nonempty set $\mathcal X$ is a symmetric function $k:\mathcal X\times\mathcal X\to\mathbb R$ such that, for every $n\geq1$, every $x_1,\ldots,x_n\in\mathcal X$, and every $c_1,\ldots,c_n\in\mathbb R$,
$$
\sum_{r,s=1}^n c_rc_s k(x_r,x_s)\geq0.
$$
Equivalently, every finite <kernel matrix> $(k(x_r,x_s))_{r,s}$ is a <positive semidefinite matrix>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 205 / 1 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-205/1/solution