How bad is Hadamard determinantal bound

Abstract

Thp Hadamard bound for t he determinant of an" by n ma trix is a good o ne in that equal it y may be a tta ined in a ri ch c lass of cases. Howe ver , th e bound gene rally g ives up a good d e al , a nd we a nswe r th e titl e ques tion " on the average." Ass uming the e ntries of A = I (J ;j ) are uniform ly di s tribut ed ove r som e interval symmetric about the origin , the expec ted va lue of the ratio of (de t A)" 10 t.h e square of the "t Hadamard bound is fo und tu be ---.: . The expecta tions of the square of the Hadama rd bound and of n1l n ! (de t A )" a re a lso computed individua ll y. a nd the ir ra t.io t urns out a lso to be . n il