How Tight is Hadamard's Bound?

Description

For a real square matrix $M$, Hadamard's inequality gives an upper bound $H$ for the determinant of $M$. This upper bound is sharp if and only if the rows of $M$ are orthogonal. In this paper 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.

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