Geometric Structures of Matrix Spaces Through Manifold Theory

Abstract

As an essential part of the modern mathematical study, matrix spaces are important to various fields (including optimization, physics, data science, and machine learning). Although conventional research on the study of the matrix spaces utilizes the approach that is largely based on the use of algebraic and linear approaches, there are numerous intrinsic properties of the matrices that are more easily observed when these spaces are viewed through a geometric lens. In this paper, the geometry of matrix spaces is explored in the context of manifold theory i.e., the use of matrices as points in differentiable manifolds of smooth types and matrices as Lie groups where present. The paper adopts a conceptual and analytic approach in examining manifold construction, tangent spaces, Riemannian metrics, geodesics, curvature and symmetry properties to the various classes of matrix spaces including; general linear, symmetric, orthogonal, and positive definite matrices. The results highlight the point that the manifold-based analysis provides a better insight into the local and global geometric behaviour and it implies that there is an overall relationship between algebraic structures and geometry. The paper also indicates the application of these geometric insights in advanced work in the area of optimization, computational mathematics, and applied areas.