I have the matrix G and I want to show that G is only positive semi definite and not positive definite. We have a vector $\mathbf{z}=(z_1,...z_n)$, along with $n$ data points $\mathbf{x}_1,...\mathbf{x}_n$. $\Phi$ is a map into dimension $d<n$ : $$z^T Gz=\langle \sum_j^n z_j\Phi(x_j)\, \sum_i^n z_i\Phi(x_j)\rangle$$
I'm thinking of finding a none zero vector $\mathbf{z}$ such that the sum inside is 0, thus giving the equality of positive semi definite. How do I find this vector?