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.