Algebraic Structure of Matrix Powers of Matrix

This paper investigates the exponential of a matrix and its inverse operation, the matrix logarithm. The matrix exponential plays a fundamental role in connecting Lie algebras with matrix Lie groups, while the logarithm provides a formal inverse in a suitable neighborhood of the identity matrix. We establish fundamental properties of these functions, construct a group structure based on generalized matrix powers, and demonstrate its isomorphism with the exponential matrix group. This research highlights the structural and algebraic significance of the exponential and logarithmic functions in the context of matrix theory.