A sign pattern that allows oppositely signed orthogonal matrices

TL;DR

This paper presents the first known example of a sign pattern matrix that admits orthogonal matrices with the same pattern but opposite determinants, addressing a question raised in prior mathematical research.

Contribution

It introduces a specific sign pattern matrix that allows for orthogonal matrices with both positive and negative determinants, expanding understanding of sign pattern constraints.

Findings

Existence of a sign pattern with orthogonal matrices of opposite determinants

First known example of such a sign pattern

Addresses a question from previous research in 1996

Abstract

We provide the first example of a sign pattern $S$ for which there exist orthogonal matrices $Q_1$ and $Q_2$ with sign pattern $S$ such that $\det Q_1=1$ and $\det Q_2=-1$. The existence of such matrices is raised by C. Waters in {"Sign Pattern Matrices That Allow Orthogonality"}, Linear Algebra and Its Applications, 235:1-13 (1996).