An ensemble of high rank matrices arising from tournaments

TL;DR

This paper investigates the rank properties of a class of symmetric matrices associated with tournaments, establishing lower bounds on their rank and showing that random matrices from this class typically have rank close to half the size.

Contribution

It introduces a subclass of matrices linked to transitive tournaments and proves they have high rank, also demonstrating that random matrices in this class generally have rank near half the matrix size.

Findings

Matrices associated with transitive tournaments have rank at least two-thirds of their size minus one.

Random matrices from the class typically have rank at least half their size with high probability.

The results connect tournament structures with matrix rank properties.

Abstract

Suppose $\mathbb{F}$ is a field and let $\mathbf{a} := (a_1, a_2, \dotsc)$ be a sequence of non-zero elements in $\mathbb{F}$. For $\mathbf{a}_n := (a_1, \dotsc, a_n)$, we consider the family $\mathcal{M}_n(\mathbf{a})$ of $n \times n$ symmetric matrices $M$ over $\mathbb{F}$ 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 $\mathcal{M}_n(\mathbf{a})$ -- which can be naturally associated with transitive tournaments -- have rank at least $\lfloor 2n/3 \rfloor - 1$. We also show that if $\operatorname{char}(\mathbb{F}) \neq 2$ and $M$ is a matrix chosen uniformly at random from $\mathcal{M}_n(\mathbf{a})$, then with high probability $\operatorname{rank}(M) \geq \bigl(\frac{1}{2} - o(1)\bigr)n$.