Submitted to ANZIAM Journal (March 8,2006) THE MATRIX EXPONENTIAL ON TIME SCALES A. ZAFER Abstract. In this paper we describe an elementary method for calculating the matrix exponential on an arbitrary time scale. An example is also given to illustrate the result. 1. Introduction There are several methods in the literature to calculate the matrix exponential e tA and the matrix A k for any given n × n matrix A due to their appearance in the study of systems of linear differential and difference equations. In 1996, I. E. Leonard [6] presented an elementary but powerful method to calculate the matrix exponential e At which uses only knowledge of homogeneous linear differential equations with constant coefficients and the Cayley-Hamilton theorem. Two years later, by using a similar approach and employing homogeneous linear difference equations with constant coefficients, M. Kwapisz [5] derived an alternative method to determine A k , the k-th power of A. A time scale T is a nonempty closed subset of the set of real numbers R. Time scale calculus, introduced by Hilger [2], has recently gained considerable interest as it unifies the continuous and discrete analysis. The most well-known examples of a time scale are T = R, T = Z, and T = q Z , where q Z = {t : t = q k ,k ∈ Z,q> 1}. In the light of time scale calculus, our aim is to develop an elementary method to calculate the matrix exponential on an arbitrary time scale, and thereby unify the results in [5, 6]. To the best of our knowledge the Putzer algorithm is the only method available to calculate the matrix exponential on an arbitrary time scale, see [1, Theorem 5.35]. Here in this article we propose an alternative method which is more elementary and easier to apply than the Putzer algorithm. For basic aspects of time scale calculus we refer to the monographs [1, 4]. In the following lines we provide only some essential ingredients to be used in this paper. 2000 Mathematics Subject Classification. 34A30. Key words and phrases. Time scale, matrix exponential, linear dynamic equation. 1