Hadamard matrices, Sequences, and Block Designs

TLDR: Seberry and Yamada as discussed by the authors considered the problem of finding the maximal determinant of real matrices with entries on the unit disc, and showed that Hadamard matrices satisfy the equality of the following inequality.

Abstract: One hundred years ago, in 1893, Jacques Hadamard [31] found square matrices of orders 12 and 20, with entries ±1, which had all their rows (and columns) pairwise orthogonal. These matrices, X = (Xij), satisfied the equality of the following inequality, |detX|2 ≤ ∏ ∑ |xij|2, and so had maximal determinant among matrices with entries ±1. Hadamard actually asked the question of finding the maximal determinant of matrices with entries on the unit disc, but his name has become associated with the question concerning real matrices. Disciplines Physical Sciences and Mathematics Publication Details Jennifer Seberry and Mieko Yamada, Hadamard matrices, Sequences, and Block Designs, Contemporary Design Theory – A Collection of Surveys, (D. J. Stinson and J. Dinitz, Eds.)), John Wiley and Sons, (1992), 431-560. This journal article is available at Research Online: http://ro.uow.edu.au/infopapers/1070 11 "-. Hadamard Matrices, Sequences, and Block Designs Jennifer Seberry and Mieko Yamada 1 IN1RODUCTION 2 HADAMARD MATRICES 3 THE SmONGEST HADAMARD CONSmUCTION THEOREMS 4 ORTIIOGONAL DESIGNS AND AsYMPTOTIC EXISTENCE 5 SEQUENCES 6 AMICABLE HADAMARD MAmICES AND AOD 7 CoNSmUCTIONS FOR SKEW HADAMARD MAmICES 8 M -SmucTUREs 9 WILLIAMSON AND WILUAMSON-TYPE MAmICES 10 SBIBD AND THE EXCESS OF HADAMARD MATRICES 11 CoMPLEX HADAMARD MATRICES APPENDIX REFERENCES