Hostname: page-component-84b7d79bbc-2l2gl Total loading time: 0 Render date: 2024-07-29T19:35:23.983Z Has data issue: false hasContentIssue false
  • English
  • Français

Equivalence classes of inverse orthogonal and unit Hadamard matrices

Published online by Cambridge University Press:  17 April 2009

R. Craigen
R. Craigen
Affiliation:
Department of Pure Mathematics, University of Waterloo Waterloo Ontario, CanadaN2L 3G1
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.

In 1867, Sylvester considered n × n matrices, (aij), with nonzero complex-valued entries, which satisfy (aij)(aij−1) = nI Such a matrix he called inverse orthogonal. If an inverse orthogonal matrix has all entries on the unit circle, it is a unit Hadamard matrix, and we have orthogonality in the usual sense. Any two inverse orthogonal (respectively, unit Hadamard) matrices are equivalent if one can be transformed into the other by a series of operations involving permutation of the rows and columns and multiplication of all the entries in any given row or column by a complex number (respectively a number on the unit circle). He stated without proof that there is exactly one equivalence class of inverse orthogonal matrices (and hence also of unit Hadamard matrices) in prime orders and that in general the number of equivalence classes is equal to the number of distinct factorisations of the order. In 1893 Hadamard showed this assertion to be false in the case of unit Hadamard matrices of non-prime order. We give the correct number of equivalence classes for each non-prime order, and orders ≤ 3, giving a complete, irredundant set of class representatives in each order ≤ 4 for both types of matrices.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 44 , Issue 1 , August 1991 , pp. 109 - 115
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Butson, A. T., ‘Generalized Hadamard matrices’, Proc. Amer. Math. Soc. 13 (1962), 894–898.CrossRefGoogle Scholar
[2]Craigen, R., ‘Embedding rectangular matrices in Hadamard matrices’ (to appear).Google Scholar
[3]de Launey, W., ‘Generalized Hadamard matrices whose rows and columns form a group’, in Combinatorial Mathematics X: Lecture Notes in Pure Mathematics 1036 (Springer-Verlag, Berlin, Heidelberg, New York, 1983).CrossRefGoogle Scholar
[4]Hadamard, J., ‘Resolution d'une question relative aux determinants’, Bull. Des Sciences Math. 17 (1893), 240–246.Google Scholar
[5]Kimura, H., ‘New Hadamard matrix of order 24’, Graphs Combin. 5 (1989), 235–242.CrossRefGoogle Scholar
[6]Seberry, J., ‘A construction for generalized Hadamard matrices’, J. Stat. Plann. Inference 4 (1980), 365–368.CrossRefGoogle Scholar
[7]Sylvester, J. J., ‘Thoughts on inverse orthogonal matrices, simultaneous sign-successions, and tessellated pavements in two or more colors, with applications to newton's rule, ornamental tile-work, and the theory of numbers’, Phil. Mag. 34 (1867), 461–475.CrossRefGoogle Scholar
[8]Wallis, W. D., Combinatorial designs: Monographs Textbooks Pure Appl. Math. 118 (Marcel Dekker, New York, 1988).Google Scholar