Geometry of Circulant Umbrella Matrices

This study provides a new perspective on the relationship between circulant matrices and umbrella matrices from the viewpoint of Lie theory, working with real-valued matrices. For the first time, we prove that orthogonal circulant matrices whose characteristic value differs from −1 are precisely umbrella matrices, and we define the associated matrix group as the Circulant Umbrella Matrix (CUM) group. Furthermore, we show that the Lie algebra of this group consists of skew-symmetric circulant matrices and we establish a direct connection between the Lie group and its Lie algebra via the Cayley transformation. To visualize these theoretical findings, we construct a four-dimensional hypersurface by selecting an appropriate parametric orbit curve and analyzing its projections in three-dimensional subspaces. In particular, we emphasize how the combination of orthogonality, circulant structure, and zero-row-sum skew-symmetry yields rich geometric and algebraic properties. Finally, we evaluate circular convolution from an orthogonality viewpoint and demonstrate that the elements of the CUM Lie group offer an alternative solution to the Yang–Baxter equation (YBE) while supporting applications in signal processing, cryptography, and other fields that depend on specialized orthogonal transformations.