Identities of matrix powers of a matrix and matrix power of matrix on lie groups

Abstract

This paper introduces fundamental definitions pertaining to Matrix Lie Groups and Lie Algebra. We illustrate the concepts of the exponential of a matrix and the logarithm of a matrix as integral components of our discourse. We introduce novel identities related to the matrix powers of a matrix. To prove these identities, we employ the results of the matrix powers of a matrix. Finally, we derive an expression for the matrix powers of a matrix within the context of a connected matrix lie group.