Additive Decompositions of Orthogonal Matrices over Finite Rings

Abstract

We investigate the problem of expressing matrices over finite rings as additive sums of orthogonal matrices. Working over the modular rings , we establish a complete positive result for all odd moduli: when , every matrix in  can be decomposed into a sum of orthogonal matrices, showing that the orthogonal group additively generates the full matrix ring. In contrast, for even moduli the situation changes drastically. We prove that in  strong parity obstructions arise: matrices with odd trace or odd off-diagonal sum cannot admit such decompositions. For we provide a full characterization, showing that a matrix is a sum of orthogonal matrices precisely when all its row sums and column sums agree in ₂. These results reveal a sharp dichotomy between odd and even moduli and illustrate how the arithmetic structure of the underlying ring governs additive decomposability into orthogonal components.