Rational Orthogonal versus Real Orthogonal

TL;DR

This paper investigates whether every real orthogonal matrix with a given zero-pattern can be approximated by a rational orthogonal matrix with the same zero-pattern, proving it for matrices of size up to 5.

Contribution

The paper formulates a conjecture relating real and rational orthogonal matrices and proves it for small matrix sizes, advancing understanding of their structural properties.

Findings

Proved the conjecture for n<=5

Established the existence of rational orthogonal matrices with the same zero-pattern as real ones for small sizes

Explored the related problem for symmetric orthogonal matrices

Abstract

The main question we raise here is the following one: given a real orthogonal n by n matrix X, is it true that there exists a rational orthogonal matrix Y having the same zero-pattern? We conjecture that this is the case and prove it for n<=5. We also consider the related problem for symmetric orthogonal matrices.