Growth in Gaussian elimination for weighing matrices, W (n, n-1)

We consider the values for large miners of a skew-Hadamard matrix or confer ence matrix W of order n and find that maximum n x n minor equals to (n - 1 )(n/2), maximum (n - 1) x (n - 1) minor equals to (n - 1)((n/2)-1), maximum (n - 2) x (n - 2) minor equals to 2(n - 1)((n/2)-2) and maximum (n - 3) x (n - 3) minor equals to 4(n - 1)((n/2)-3). This leads us to conjecture that the growth factor for Gaussian elimination (GE) of completely pivoted (CP) skew-Hadamard or conference matrices and indeed any CP weighing matrix of order n and weight n - 1 is n - 1 and that the first and last few pivots ar e (1, 2, 2, 3 or 4,..., n - 1 or (n - 1)/2, (n - 1)/2, n - 1) for n > 14, W e show the unique W(6, 5) has a single pivot pattern and the unique W(8, 7) has at least two pivot structures. We give two pivot patterns for the uniq ue W(10, 9), (C) 2000 Elsevier Science Inc. All rights reserved.