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The growth factor of a Hadamard matrix of order 16 is 16
Author(s) -
Kravvaritis Christos,
Mitrouli Marilena
Publication year - 2009
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.637
Subject(s) - mathematics , hadamard matrix , hadamard transform , hadamard's maximal determinant problem , block matrix , matrix (chemical analysis) , combinatorics , complex hadamard matrix , discrete mathematics , eigenvalues and eigenvectors , mathematical analysis , physics , materials science , quantum mechanics , composite material
In 1968 Cryer conjectured that the growth factor of an n × n Hadamard matrix is n . In 1988 Day and Peterson proved this only for the Hadamard–Sylvester class. In 1995 Edelman and Mascarenhas proved that the growth factor of a Hadamard matrix of order 12 is 12. In the present paper we demonstrate the pivot structures of a Hadamard matrix of order 16 and prove for the first time that its growth factor is 16. The study is divided in two parts: we calculate pivots from the beginning and pivots from the end of the pivot pattern. For the first part we develop counting techniques based on symbolic manipulation for specifying the existence or non‐existence of specific submatrices inside the first rows of a Hadamard matrix, and so we can calculate values of principal minors. For the second part we exploit sophisticated numerical techniques that facilitate the computations of all possible ( n − j ) × ( n − j ) minors of Hadamard matrices for various values of j . The pivot patterns are obtained by utilizing appropriately the fact that the pivots appearing after the application of Gaussian elimination on a completely pivoted matrix are given as quotients of principal minors of the matrix. Copyright © 2009 John Wiley & Sons, Ltd.