How Tight is Hadamard's Bound?

CONTENTS For a real square matrix M, Hadamard's inequality gives an up1 . Introduction p e r bound H for the determinant of M; the bound is sharp if and 2. Distribution only if the rows of M are orthogonal. We study how much we 3. Analysis of |d|/H can expect that H overshoots the determinant of M, when the 4. Analysis of Iog(|d|/H) r o w s °f M are chosen randomly on the surface of the sphere. 5 Experiments ^ i s 8 l ves a n indication of the "wasted effort" in some modular r ~ , . algorithms. 6. Conclusions &