Abstract
We consider the values for large minors of a skew-Hadamard matrix or conference matrix W of order n and find 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 of completely pivoted skew-Hadamard or conference matrices and indeed any completely pivoted 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 or 4, ..., 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(12, 11).
Publication Details
This article was originally published as Koukouvinos, C, Mitrouli, M and Seberry, J, Growth in Gaussian Elimination for Weighing Matrices, W (n, n — 1), Linear and Multilinear Algebra, 306, 2000, 189-202. Copyright Taylor & Francis. Original journal available here.