How Good is Hadamard’s Inequality for Determinants?

Abstract

AbstractLet A be a real n × n matrix and define the Hadamard ratio h(A) to be the absolute value of det A divided by the product of the Euclidean norms of the columns of A. It is shown that if A is a random variable whose distribution satisfies some simple symmetry properties then the random variable log h(A) has mean and variance . In particular, for each ε > 0, the probability that h(A) lies in the range tends to 1 as n tends to ∞.