Electronic Journal of Differential Equations,
Vol. 2014 (2014), No. 79, pp. 1-7.

Title: Explicit expressions for the  matrix exponential function 
       obtained by means of an algebraic convolution formula

Authors: Jose Roberto Mandujano (Univ. Zacatenco,  Mexico D.F., Mexico)
         Luis Verde-Star (Univ. Autonoma Metropolitana, Mexico D.F., Mexico)

Abstract:
 We present a direct algebraic method for obtaining  the matrix exponential
 function  exp(tA), expressed  as an explicit function of t for any square
 matrix A whose eigenvalues are known. The explicit form can be used to
 determine how a given eigenvalue affects the behavior of  exp(tA).
 We use an algebraic convolution formula for basic exponential polynomials
 to obtain the dynamic solution g(t) associated with the characteristic
 (or minimal) polynomial w(x) of A.
 Then  exp(tA) is  expressed as $\sum_k g_k(t) w_k(A)$, where the
 $g_k(t)$ are successive derivatives of g and the $w_k$ are the
 Horner polynomials associated with w(x).
 Our method also gives a number $\delta$  that measures how well the computed
 approximation satisfies the matrix differential equation
 F'(tA)=A F(tA) at some given point $t=\beta$.
 Some numerical experiments with random matrices  indicate that the proposed
 method  can be applied to matrices of order up to 40.

Submitted June 20, 2013. Published March 21, 2014.
Math Subject Classifications: 34A30, 65F60, 15A16.
Key Words: Matrix exponential; dynamic solutions; explicit formula;
           systems of linear  differential equations.