Positive Semidefinite Kernel on a Finite Set

Let $\mathbb{C}$ be the field of complex numbers, containing the real numbers, with conjugation $z\mapsto\overline{z}$. Let $F$ be a nonempty finite set; the set $F\times F$ is nonempty and finite by claim 1 of Finiteness of Cartesian Products, Tuple Sets, and Permutation Sets, and sums over it are those of Sum over a Finite Index Set.

1. (Kernel)¶ A map $K:F\times F\to\mathbb{C}$ is a positive semidefinite kernel on $F$ if $K(v,u)=\overline{K(u,v)}$ for all $u,v\in F$ and, for every map $z:F\to\mathbb{C}$, the number

is real and nonnegative. The number $Q_{K}(z)$ is the quadratic form of $K$ at $z$.