More explicit formulas for the matrix exponential

The matrix exponential plays an important role in the study of dynamical systems and linear systems. An elementary method is discussed for computing etA for general . It is shown that this method can be applied to generate explicit formulas for etA in various forms. Using this method, some well-known formulas are rederived, and some new formulas are also derived. As applications, explicit formulas for n = 3 or n = 4 are given. In each case, a characterization of etA is also made, based on the entries of A alone.