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

Mandujano, Jose Roberto ; Verde-Star, Luis

Electronic journal of differential equations, Tome 2014 (2014)

Résumé

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.

Détail

Classification : 34A30, 65F60, 15A16
Keywords: matrix exponential, dynamic solutions, explicit formula, systems of linear differential equations