Gaussian kernel (SVM)

The gaussian kernel is a function that can be used in the Kernel trick for SVMs. It has one variable $\sigma \in \mathbb{R}$ with $\sigma \not = 0$. It is defined as

$$K_{\sigma}: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}, \mbox{ by } K_{\sigma}(x_1, x_2) = \exp \left ( - \frac{\vert \vert x - y \vert \vert^2}{2 \sigma^2} \right ).$$