 Solution

For every $m\ge 1$, every choice $x_1,\dots ,x_m\in \mathcal{X}$, and every coefficient vector $\alpha \in {\mathbb{R}}^m$, the required condition isThus ${\alpha }^{T}K\alpha \ge 0$ for the kernel matrix $K$. For a feature map, this follows fromIn kernel methods, the term positive-definite kernel commonly uses this non-strict inequality. Strict positive definiteness is the stronger requirement that the expression be positive for distinct inputs and every nonzero $\alpha$; it is not required here.