Positive Semidefinite Kernels on a Finite Set: Rank-One Decomposition and the Schur Product

Let $\mathbb{C}$ be the field of complex numbers, containing the real numbers, with conjugation $z\mapsto\overline{z}$. Let $\mathbb{N}$ be the set of natural numbers. Let $F$ be a nonempty finite set, and let positive semidefinite kernels on $F$ and their quadratic forms $Q_{K}$ be those of Positive Semidefinite Kernel on a Finite Set §kernel; sums over a finite index set are those of Sum over a Finite Index Set.

1. (Sums of rank-one kernels)¶ Let $N\in\mathbb{N}$ and let $b_{1},\dots,b_{N}:F\to\mathbb{C}$ be maps. Then $K(u,v)=\sum_{k=1}^{N}\overline{b_{k}(u)}\,b_{k}(v)$ defines a positive semidefinite kernel on $F$.

2. (Rank-one decomposition)¶ If $F$ has $N$ elements and $K$ is a positive semidefinite kernel on $F$, then there are maps $b_{1},\dots,b_{N}:F\to\mathbb{C}$ with $K(u,v)=\sum_{k=1}^{N}\overline{b_{k}(u)}\,b_{k}(v)$ for all $u,v\in F$.

3. (Schur product)¶ If $K$ and $L$ are positive semidefinite kernels on $F$, then the map $(u,v)\mapsto K(u,v)L(u,v)$ is a positive semidefinite kernel on $F$.