I have $n$ linearly inseperable datapoints, $x_1 \dots , x_n$. I use the kernel trick to map and compute the dot product in higher dimensions (without actually mapping / transforming the data).
Assume I used the kernel $k(x,y).$ The entry at position $i,j$ of an associated "kernel matrix" is defined as $k(x_i,x_j).$ This matrix is apparently also known as the "Gram" matrix.
What is this matrix used for? Why do I need it? This is in the context of support vector machines, if it matters.