Positive-definite kernel

Remarks

Such a kernel is Hermitian: $K(y, x) = \overline{K(x, y)}$, because each matrix $\big(K(x_i, x_j)\big)$ is a Hermitian matrix. For a real-valued kernel one often uses real coefficients $c_i$ and asks in addition that $K(x, y) = K(y, x)$. Some authors call these kernels positive semidefinite and keep "positive definite" for the case of strict inequality when the points are distinct and $c \ne 0$.

Example: if $\Phi \colon X \to \mathcal{H}$ is any map into a Hilbert space, then $K(x, y) = \ip{\Phi(x)}{\Phi(y)}$ is positive-definite, where the inner product is linear in the first argument. Indeed $\sum_{i,j} c_i \overline{c_j} \ip{\Phi(x_i)}{\Phi(x_j)} = \norm{\sum_i c_i \Phi(x_i)}^2 \ge 0$. By the Moore–Aronszajn theorem every positive-definite kernel arises in this way, from its reproducing kernel Hilbert space.

Non-example: the constant kernel $K(x, y) = -1$ is not positive-definite. With $n = 1$ and $c_1 = 1$, the sum is $K(x_1, x_1) = -1 < 0$.

If $X = G$ is a group and $K(g, h) = \varphi(g^{-1}h)$, then $K$ is positive-definite exactly when $\varphi$ is a positive-definite function on $G$.