Symmetric and skew-symmetric $\{0,\pm 1\}$-matrices with large determinants

TL;DR

This paper explores the relationship between $\pm 1 ext{-matrices}$ with maximal determinants and tournament matrices, proving a conjecture and analyzing spectral properties of conference matrices.

Contribution

It establishes the equivalence between maximal determinant $\pm 1 ext{-matrices}$ and tournament matrices, confirming a recent conjecture and characterizing large submatrices of conference matrices.

Findings

Maximal determinant $\pm 1 ext{-matrices}$ are equivalent to certain tournament matrices.

Confirmed a recent conjecture by Armario.

Large submatrices of conference matrices are uniquely determined by their spectrum.

Abstract

We show that the existence of $\{\pm 1\}$-matrices having largest possible determinant is equivalent to the existence of certain tournament matrices. In particular, we prove a recent conjecture of Armario. We also show that large submatrices of conference matrices are determined by their spectrum.