2024/10/15 by Q.W Tang, Wei Wang, Tang, Quanyu +3 · 1 citation
Computer Science · #05C50 #Combinatorics (math.CO) #FOS: Mathematics #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2410.11246
openalex publication_date 2024/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 = (In + S)-1(In - S)P, where In 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 holds 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.