Kernel

feature mapping $\phi : R^n \rightarrow R^t$

replace $x^{(i)}$ with $\phi(x^{(i)})$

since $X=[x^{(i)T}] \epsilon R^{m*1}$, $\phi = [\phi(x^{(i)T})] \epsilon R^{m*1}$

$K_{ij}=\phi(x^{(i)})^T\phi(x^{(j)}) \epsilon R^{m*m} = \phi \phi^T \epsilon R^{m*m}$