Low-rank matrices, tournaments, and symmetric designs

Let a=(a(i))(i >= 1) be a sequence in a field F, and f:F x F -> F be a function such that f(a(i),a(i))not equal 0 for all i >= 1. For any tournament T over [n], consider the nxn symmetric matrix MT with zero diagonal whose (i,j)th entry (for i<j) is f(a(i),a(j)) if i -> j in T, and f(a(j),a(i)) if j -> i in T. It is known (cf. Balachandran et al., Linear Algebra Appl. 658 (2023), 310-318) that if T is a uniformly random tournament over [n], then rank(M-T)>=((1)/(2)-o(1))n with high probability when char(F)not equal 2 and f is a linear function. In this paper, we investigate the other extremal question: how low can the ranks of such matrices be? We work with sequences a that take only two distinct values, so the rank of any such n x n matrix is at least n/2. First, we show that the rank of any such matrix depends on whether an associated bipartite graph has certain eigenvalues of high multiplicity. Using this, we show that if f is linear, then there are real matrices M-T(f;a) of rank at most (n)/(2)+O(1). For rational matrices, we show that for each epsilon > 0 we can find a sequence a(epsilon) for which there are matrices M-T(f;a) of rank at most (12+epsilon)n+O(1). These matrices are constructed from symmetric designs, and we also use them to produce bisection-closed families of size greater than left perpendicular3n/2right perpendicular-2 for n <= 15, which improves the previously best known bound given in [5]. (c) 2024 Elsevier Inc. All rights reserved.