An ensemble of high rank matrices arising from tournaments

Abstract

Suppose F is a field and let a := (a 1 , a 2 , . . . ) be a sequence of non-zero elements in F. For a n := (a 1 , . . . , a n ), we consider the family M n (a) of n × n symmetric matrices M over F with all diagonal entries zero and the (i, j)th element of M either a i or a j for i < j. In this short paper, we show that all matrices in a certain subclass of M n (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 M n (a), then with high probability rank(M ) ≥ 1 2 − o(1) n.