The Hadamard Operator Norm of a Circulant and Applications

Abstract

Let $M_n $ be the space of $n \times n$ complex matrices and let $\| \cdot \|_\infty $ denote the spectral norm. Given matrices $A = [ a_{ij} ]$ and $B = [ b_{ij} ]$ of the same size, define their Hadamard product to be $A \circ B = [ a_{ij} b_{ij} ]$. Define the Hadamard operator norm of $A \in M_n $ to be \[ \| | A | \|_\infty = \max \{ \| A \circ B \|_\infty : \| B \|_\infty \leq 1 \}. \] It is shown that \[ (1) \qquad |||A|||_\infty = {\operatorname{tr}} | A |/n \] if and only if \[ (2) \qquad | A | \circ I = | A^ * | \circ I = ( {\operatorname{tr}} | A |/n ) I. \] 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 \[ |||{\operatorname{sign}} ( j - i )_{i,j = 1}^n |||_\infty ,\quad ||| [ ( \lambda _i - \lambda _j )/( \lambda _i + \lambda _j ) ]_{i, j = 1}^n |||_\infty, \quad |||T_n |||_\infty, \] where $T_n $ is the $n \times n$ 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 \[ \sup \{ \| \,| A | - | B |\, \|_\infty : \| A - B \|_\infty \leq 1,A,B \in M_n \} \] and \[ \sup \{ \| \, | A | B - B | A |\, \|_\infty : \| AB - BA \|_\infty \leq 1,A,B \in M_n , A = A^ * \}. \] 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.