How Tight is Hadamard's Bound?

Abstract

For a real square matrix M, Hadamard's inequality gives an upper bound H for the determinant of M; the bound is sharp if and only if the rows of M are orthogonal. We study how much we can expect that H overshoots the determinant of M, when the rows of M are chosen randomly on the surface of the sphere. This gives an indication of the “wasted effort” in some modular algorithms.