# What is the kernel matrix used for in the kernel trick?

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.