The Generation of All Regular Rational Orthogonal Matrices

Abstract: A \emph{rational orthogonal matrix} $Q$ is an orthogonal matrix with rational entries, and $Q$ is called \emph{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 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 $-1$.