Some Explicit Formulas for Matrix Exponential, Matrix Logarithm, the $n$th Power of Matrices and their Drazin Inverses

TL;DR

This paper introduces new explicit formulas for matrix exponential, logarithm, powers, and Drazin inverses, providing a direct, elementary approach that generalizes and recovers existing results, with practical examples.

Contribution

It offers novel, manageable closed-form formulas for matrix functions and inverses, including generalizations of classical decompositions and spectral projections.

Findings

New formulas for matrix exponential and logarithm

Explicit expressions for matrix powers and Drazin inverses

Illustrative examples demonstrating the formulas' applications

Abstract

In this work, new closed-form formulas for the matrix exponential are provided. Our method is direct and elementary, it gives tractable and manageable formulas not current in the extensive literature on this essential subject. Moreover, others are recuperated and generalized. As a consequence, we easily obtain the Chevalley{Jordan decomposition and the spectral projections of any matrix. In addition, closed-form expressions for the arbitrary positive powers of matrices and their Drazin inverses are presented. Using these results, an elegant explicit formula for logarithm of matrices is obtained. Several particular cases and examples are formulated to illustrate the methods presented in this paper.