The Matrix Exponential

The exponential of a constant matrix on time scales

“…Zafer [6] From the general solution one can easily get the special solutions Jt,-(f)> ' = 0, 1,2, satisfying (2.7). Indeed, it turns out that f a 2 If we take T = K, t 0 = 0, a = 2 and b = 3, then our example coincides with the one given by Leonard [7].…”

“…In 1996,1. E. Leonard [7] presented an elementary but powerful method to calculate the matrix exponential e M 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 [6] derived an alternative method to determine A k , the k-th power of A.…”

“…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 100 A. Zafer [2] results in [6,7]. To the best of our knowledge the Putzer algorithm is the only method available to calculate the exponential of a constant matrix on an arbitrary time scale, see [1,Theorem 5.35].…”

Matrix Exponentials---Another Approach

“…Others (cf. Ziebur (1970), Fulmer (1975, Leonard (1996), and Elaydi and Harris (1998)) have discussed portions of the method described here, but these references did not develop this method with the ease of use shown to us by Harris.…”

Integers Powers of Certain Asymmetric Matrices

“…The recurrence method, which is an alternative to the Jordan's decomposition method, Leonard's algorithm, Putzer's algorithm and the method of characteristic polynomial [1,2,3,4,5,6,7], seems fast and facilitating the definition of integers powers of any matrices. The paper is a follow up on [8] (which is inspired by [9,10], see also [11]), where the cases of the third and fourth order asymmetric matrices are discussed and where also the complex powers of these matrices are defined.…”