Number Theory pp 223-259 | Cite as

Hadamard’s Determinant Problem

  • W. A. Coppel
Part of the Universitext book series (UTX)


It was shown by Hadamard (1893) that, if all elements of an n × n matrix of complex numbers have absolute value at most μ, then the determinant of the matrix has absolute value at most μ n n n/2. For each positive integer n there exist complex n × 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, kn), 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 × n matrix, where n > 2, then n is divisible by 4.


Simple Group Incidence Matrix Hadamard Matrice Hadamard Matrix Real Symmetric Matrix 
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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • W. A. Coppel
    • 1
  1. 1.GriffithAustralia

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