Generalized tournament matrices with the same principal minors

TL;DR

This paper characterizes generalized tournament matrices with identical principal minors of orders 2, 3, and 4, showing these minors determine all other principal minors, thus revealing their structural significance.

Contribution

It provides a complete characterization of generalized tournament matrices based on principal minors of small orders, establishing their determining role for the entire matrix structure.

Findings

Principal minors of orders 2, 3, and 4 determine all other principal minors.

A characterization of generalized tournament matrices with equal principal minors is established.

The results reveal the structural importance of small-order principal minors in these matrices.

Abstract

A generalized tournament matrix $M$ is a nonnegative matrix that satisfies $M+M^{t}=J-I$, where $J$ is the all ones matrix and $I$ is the identity matrix. In this paper, a characterization of generalized tournament matrices with the same principal minors of orders $2$, $3$, and $4$ is given. In particular, it is proven that the principal minors of orders $2$, $3$, and $4$ determine the rest of the principal minors.