V Hadamard’s Determinant Problem.............................
It was shown by Hadamard (1893) that, if all elements of ann×nmatrix of complex
numbers have absolute value at mostμ, then the determinant of the matrix has absolute
value at mostμnnn/^2. For each positive integernthere exist complexn×nmatrices
for which this upper bound is attained. For example, the upper bound is attained for
μ=1 by the matrix(ωjk)( 1 ≤j,k≤n),whereωis a primitiven-th root of unity.
This matrix is real forn= 1 ,2. However, Hadamard also showed that if the upper
bound is attained for a realn×nmatrix, wheren>2, thennis divisible by 4.
Without loss of generality one may supposeμ=1. A realn×nmatrix for which
the upper boundnn/^2 is attained in this case is today called aHadamard matrix.It
is still an open question whether ann×nHadamard matrix exists for every positive
integerndivisible by 4.
Hadamard’s inequality played an important role in the theory of linear integral
equations created by Fredholm (1900), and partly for this reason many proofs and
generalizations were soon given. Fredholm’s approach to linear integral equations has
been superseded, but Hadamard’s inequality has found connections with several other
branches of mathematics, such as number theory, combinatorics and group theory.
Hadamard matrices have been used to enhance the precision of spectrometers, to
design agricultural experiments and to correct errors in messages transmitted by
spacecraft.
The moral is that a good mathematical problem will in time find applications.
Although the case wherenis divisible by 4 has a richer theory, we will also treat
other cases of Hadamard’s determinant problem, since progress with them might lead
to progress also for Hadamard matrices.
1 WhatisaDeterminant?
The system of two simultaneous linear equations
α 11 ξ 1 +α 12 ξ 2 =β 1
α 21 ξ 1 +α 22 ξ 2 =β 2W.A. Coppel, Number Theory: An Introduction to Mathematics, Universitext, 223
DOI: 10.1007/978-0-387-89486-7_5, © Springer Science + Business Media, LLC 2009