The growth factor of a Hadamard matrix of order 16 is 16

Abstract

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.

Anymatrix: an extensible MATLAB matrix collection

“…Cryer [9, Conjecture 5.1] conjectured that ρ n = n for any Hadamard matrix. This conjecture is still open [40,44] and substantial effort was required even to show that ρ 16 = 16 for any 16 × 16 Hadamard matrix [39]. We will check the value of ρ n for the real Hadamard matrices in Anymatrix using the following code.…”

Determinantal Properties of Generalized Circulant Hadamard Matrices

“…More specifically, the first 5 and the last 4 pivots for any Hadamard matrix were determined. This effort allowed to characterize the growth of Hadamard matrices up to order 16 [11], but still left the Cryer's Conjecture unsolved. With regard to these facts, it is natural to examine the growth factor on matrix classes that are related to Hadamard matrices.…”

Gaussian elimination

“…This was finally resolved by Gould10 and Edelman11, who found an example with ρ 13 > 13. Research on certain aspects of the size of ρ n for complete pivoting is ongoing 12. Interestingly, ρ n ≥ n for any Hadamard matrix (a matrix of 1's and −1's with orthogonal columns) and any pivoting strategy 13.…”