Operator Identities and the Solution of Linear Matrix Difference and Differential Equations

Abstract

We use operator identities in order to solve linear homogeneous matrix differenee and differential equations and we obtain several explicit formulas for the exponential and for the powers of a matrix as an example of our methods. Using divided differenees we find solutions of some sealar initial value problems and we show how the solution of matrix equations is related to polynomial interpolation.

Fibonacci-Horner decomposition of the matrix exponential and the fundamental system of solutions

“…Moreover, using (2.6), we establish easily that the k-th derivative of the function ϕ(t) satisfies ϕ (k) (0) = 0 for k = 0, 1, · · · , r − 2 and ϕ (r−1) (0) = 1. Hence, the function ϕ(t) given by (2.6) is nothing else but the dynamical solution of the preceding differential equations (see [7,13]). Also (2.5) shows that the coefficients of e tA in the Fibonacci-Horner basis are the elements of the fundamental system of solutions {ϕ(t), ϕ (t), · · · , ϕ (r−2) (t), ϕ (r−1) (t)} of the preceding differential equation.…”

“…Remark 2.3. In [13] improved expressions for A n and e tA are obtained with the aid of the dynamical solution. In the latter case, this is done in various forms, using a specific technique of the theory of divided differences.…”

“…Meanwhile, recently Verde-Star's theory permits to obtain the polynomials R j (t) (1 ≤ j ≤ s) using the method of divided difference (see [13]). In Subsection 4.2 we bring focus to this observation.…”

“…2)-(3.6) of [13] for more details). More precisely, the result of Theorem 4.2 is expressed in Corollary 4.2 of [13] in terms of the functionals L k,j .…”

“…There is a large variety of methods for computing the exponential of a matrix and the differential equation method is among them; see [4,6,7,9,12,13] and references therein. As a consequence, diverse decompositions of e tA have been considered in the literature, associated with various kinds of bases for the vector space of polynomials; see, e.g., [4,6,7,9,12,13,14]. An approach for the decomposition of e tA related to the combinatorial expression of generalized Fibonacci sequences (see [8,10]) is studied in the recent papers [1,2,11].…”