How can we take use Cholesky factorization to show that a given symmetric matrix, M is positive semi-definite?

Question

Let's discard the fact that we know anything about the eigenvalues of being real and positive. Can we still prove a symmetric matrix to be semi-definite? What should be its properties? I have a matrix, M of the form M[i,j] = 1/n*$\sum_{k=1}^n x[k+i]x[k+j]$, how can I prove this to be a PSD?