Back Matter

Abstract

In general,is only true, when the matrices X and Y commute (i.e., XY = YX). As noted previously, the solution in Eq. (2.2) for the system in Eq. (2.1) is only valid when the matrices S and A d commute, and in general they do not (see Eqs. (2.9) and (2.10)). Here it is shown that when the matrices A d and A in Eq. (2.1) commute, then S and A will commute, and the solution in Eq. (2.2) becomes valid. From Eq. (2.14) it is noted that S can be expressed in terms of a polynomial function of the matrices A d and A, since both the exponential and Lambert W functions are represented as such polynomial series (Corless et al., 1996). In general if two matrices X and Y commute, and the matrix functions f(X) and g(Y) can be expressed in a polynomial series form, i.e.,where p k and q k are arbitrary coefficients, then (Pease, 1965) For the last equation in Eq. (A.5) to hold, for any value of t ∈ [0, h], one can conclude as