ON TOPOLOGY OF CENTROSYMMETRIC MATRICES WITH APPLICATIONS

In this work, we investigate the algebraic and geometric properties of centrosymmetric matrices over the positive reals. We show that the set of centrosymmetric matrices, denoted as $\mathcal{C}_n$, forms a Lie algebra under the Hadamard product with the Lie bracket defined as $[A, B] = A \circ B - B \circ A$. Furthermore, we prove that the set $\mathcal{C}_n$ of centrosymmetric matrices over $\mathbb{R}^+$ is an open connected differentiable manifold with dimension $\lceil \frac{n^2}{2}\rceil$. This result is achieved by establishing a diffeomorphism between $\mathcal{C}_n$ and a Euclidean space $\mathbb{R}^{\lceil \frac{n^2}{2}\rceil}$, and by demonstrating that the set is both open and path-connected. This work provides insight into the algebraic and topological properties of centrosymmetric matrices, paving the way for potential applications in various mathematical and engineering fields.