Hadamard’s Determinant Problem

Abstract

It was shown by Hadamard (1893) that, if all elements of an n {\texttimes} n matrix of complex numbers have absolute value at most $μ$, then the determinant of the matrix has absolute value at most $μ$nnn/2. For each positive integer n there exist complex n {\texttimes} n matrices for which this upper bound is attained. For example, the upper bound is attained for $μ$ = 1 by the matrix ($ω$jk)(1 ≤ j, k ≤ n), where $ω$ is a primitive n-th root of unity. This matrix is real for n = 1, 2. However, Hadamard also showed that if the upper bound is attained for a real n {\texttimes} n matrix, where n > 2, then n is divisible by 4.