Growth in Gaussian elimination for weighing matrices, W(n,n−1) Original Research Article

We consider the values for large minors of a skew-Hadamard matrix or conference matrix W of order n and find that maximum n×n minor equals to (n−1)n/2, maximum (n−1)×(n−1) minor equals to (n−1)(n/2)−1, maximum (n−2)×(n−2) minor equals to 2(n−1)(n/2)−2, and maximum (n−3)×(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 are (1,2,2,3 or4,…,n−1 or (n−1)/2,(n−1)/2,n−1) for n>14. We 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 unique W(10,9).