Let M n be the space of n n complex matrices and let k k 1 denote the spectral norm. Given matrices A = a ij ] and B = b ij ] of the same size we deene their Hadamard product to be A B = a ij b ij ]. We deene the Hadamard operator norm of A 2 M n by j j jAj j j 1 = maxfkA Bk 1 : kBk 1 1g: We show that j j jAj j j 1 = tr jAj=n (1) if and only if jAj I = jA j I = (tr jAj=n)I: (2) We show that (2) holds for generalized circulants and hence that the Hadamard operator norm of a generalized circulant can be computed easily. This allows us to compute or bound j j jsign(j ? i)] n i;j=1 j j j 1 ; j j j(i ? j)=(i + j)] n i;j=1 j j j 1 ; j j jT n j j j 1 (where T n is the n n matrix with ones on and above the diagonal and zeros below) and related quantities. In each case the norms grow like log n. Using these results we obtain upper and lower bounds on quantities of the form g We also indicate the extent to which our results generalize to all unitarily invariant norms, characterize the case of equality in a matrix Cauchy-Schwarz Inequality, and give a counterexample to a conjecture involving Hadamard products.