Generating all regular rational orthogonal matrices

Abstract

A rational orthogonal matrix Q is an orthogonal matrix with rational entries, and Q is called regular if each of its row sum equals one, i.e., $Qe=e$ where e is the all-one vector. This paper presents a method for generating all regular rational orthogonal matrices using the classic Cayley transformation. Specifically, we demonstrate that for any regular rational orthogonal matrix Q, there exists a permutation matrix P such that QP does not possess an eigenvalue of −1. Consequently, Q can be expressed in the form $Q={(I_n+S)}^{-1}(I_n-S)P$, where $I_n$ is the identity matrix of order n, S is a rational skew-symmetric matrix satisfying $Se=0$, and P is a permutation matrix. Central to our approach is a pivotal intermediate result, which is of independent interest: given a square matrix M, then MP has −1 as an eigenvalue for every permutation matrix P if and only if either every row sum of M is −1 or every column sum of M is −1.