Low-rank matrices, tournaments, and symmetric designs

Abstract

Let a=(ai)i≥1 be a sequence in a field F, and f:F×F→F be a function such that f(ai,ai)≠0 for all i≥1. For any tournament T over [n], consider the n×n symmetric matrix MT with zero diagonal whose (i,j)th entry (for i<j) is f(ai,aj) if i→j in T, and f(aj,ai) if j→i in T. It is known [3] that if T is a uniformly random tournament over [n], then rank(MT)≥(12−o(1))n with high probability when char(F)≠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×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 MT(f;a) of rank at most n2+O(1). For rational matrices, we show that for each ε>0 we can find a sequence a(ε) for which there are matrices MT(f;a) of rank at most (12+ε)n+O(1). These matrices are constructed from symmetric designs, and we also use them to produce bisection-closed families of size greater than ⌊3n/2⌋−2 for n≤15, which improves the previously best known bound given in [5].