Consider any Mercer kernel defined by k(x, x) = (x) >(x). We are given a sample S = {x1, x2, ..., xn} of n inputs. We can form the Kernel (Gram) matrix K as an n n matrix of kernel evaluations between all pairs of examples i.e., Ki,j = k(xi , xj ). Mercer's Theorem states that a symmetric function k(., .) is a kernel iff for any finite sample S the kernel matrix K is positive semi-definite. Recall that a matrix K R nn is positive semi-definite iff c >Kc 0 for all real-valued vectors c R n. Prove Mercer's theorem in one direction: for any Mercer kernel k(., .) and finite sample S, the kernel matrix K is positive semi-definite