Hostname: page-component-cb9f654ff-9b74x Total loading time: 0 Render date: 2025-08-20T03:51:31.551Z Has data issue: false hasContentIssue false
  • English
  • Français

Hurwitz on Hadamard designs

Published online by Cambridge University Press:  17 April 2009

T. Storer
T. Storer
Affiliation:
University of Michigan, Ann Arbor, Michigan, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An n × n–matrix on n signed variables is called Hadamard of Williamson type if each variable occurs exactly once in each row, and the inner product of any pair of distinct rows is zero. We show here that these matrices correspond in a natural way to rational formulas for products of sums of n squares, shown by Hurwitz to exist only for n = 1, 2, 4, and 8. Hurwitz' arguments contain an implicit proof that this correspondence is one-to-one (we show this directly) and hence that Hadamard matrices of Williamson type exist for orders 1, 2, 4 and 8 only.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 4 , Issue 1 , February 1971 , pp. 109 - 112
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Baumert, L.D. and Hall, Marshall Jr, “A new construction for Hadamard matrices”, Bull. Amer. Math. Soc. 71 (1965), 169–170.CrossRefGoogle Scholar
[2]Hall, Marshall Jr, Combinatorial theory (Blaisdell Publishing Co. [Ginn and Co.], Waltham, Massachusetts; Toronto, Ontario; London; 1967).Google Scholar
[3]Hurwitz, Adolf, “Über die Komposition der quadratischen Formen von beliebig vielen Variabeln”, Nachr. Königl. Ges. Wiss. Göttingen, Math.-phys. Klasse (1898), 309–316; Math. Werke, Band 2, 565–571 (Birkhäuser, Basel, 1933).Google Scholar
[4]Hurwitz, A., “Über die Komposition der quadratischen Formen”, Math. Ann. 88 (1923), 1–25; Math. Werke, Band 2, 641–666 (Birkhäuser, Basel, 1933).CrossRefGoogle Scholar
[5]Wallis, Jennifer, “Hadamard matrices”, Bull. Austral. Math. Soc. 2 (1970), 45–54.Google Scholar
[6]Williamson, John, “Hadamard's determinant theorem and the sum of four squares”, Duke J. Math. 11 (1944), 65–81.CrossRefGoogle Scholar