An ensemble of high rank matrices arising from tournaments

Abstract

Suppose F is a field and let a≔(a1,a2,…) be a sequence of non-zero elements in F. For an≔(a1,…,an), we consider the family Mn(a) of n×n symmetric matrices M over F with all diagonal entries zero and the (i,j)th element of M either ai or aj for i<j. In this short paper, we show that all matrices in a certain subclass of Mn(a)—which can be naturally associated with transitive tournaments—have rank at least ⌊2n/3⌋−1. We also show that if char(F)≠2 and M is a matrix chosen uniformly at random from Mn(a), then with high probability rank(M)≥(12−o(1))n.