Decomposition of ω-circulant matrices over quaternion algebra

Complex ω-circulant matrices are diagonalizable by combining the discrete Fourier transform with a diagonal scaling determined by ω. This simple factorization is a basic tool in the analysis and fast solution of structured linear systems. Over the quaternion algebra, however, noncommutativity prevents a direct transfer of the complex argument. In this paper, we identify the precise commutativity condition required for the diagonal scaling to remain valid, under which a quaternion ω̆-circulant matrix can still be reduced to a quaternion circulant matrix. Combining this reduction with the block-diagonalization of quaternion circulant matrices yields a diagonal-plus-anti-diagonal decomposition. The result clarifies precisely which part of the classical ω-circulant theory is valid in the quaternion setting.