THE HADAMARD OPERATOR NORM OF A CIRCULANT AND APPLICATIONS

Let Mn be the space of n x n complex matrices and let \\.\\infinity denote the spectral norm. Given matrices A = [a(ij)] and B = [b(ij)] of the same size, define their Hadamard product to be A . B = [a(ij)b(il)]. Define the Hadamard operator norm of A is-an-element-of M(n) to be \\\A\\\infinity = max {\\A . B\\infinity : \\B\\infinity less-than-or-equal-to 1}. It is shown that \\A\\infinity = tr\A\/n if and only if \A\ . I = \A*\ . I = (tr\A\/n)I. It is shown 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 \\\[sign(j - i)n/i,j=1\\\infinity, \\\[lambda(i) - lambda(j))/(lambda(i) + lambda(j)]n/i,j=1\\\infinity, \\\T(n)\\\infinity, where T(n) is the n x 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 upper and lower bounds are obtained on quantities of the form sup{\\ \A\ - \B\ \\infinity : \\A - B\\infinity less-than-or-equal-to 1, A, B, is-an-element-of M(n)} and sup{\\ \A\B - B\A\ \\infinity : \\AB - BA\\infinity less-than-or-equal-to 1, A, B is-an-element-of M(n), A = A*} The authors also indicate the extent to which the 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.