Some Explicit Formulas for Matrix Exponential, Matrix Logarithm, the th Power of Matrices and their Drazin Inverses
arXiv:2010.12551
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.
23 pages; the abstract is changed; corrected typos; some examples are added; some references are omitted and others are added or modified; some remarks are omitted