Finding Matrices that Satisfy Functional Equations

Abstract

At first glance, this problem appears to be quite difficult. Beyond the likely difficulty of finding such a matrix N(x), it is not even immediately clear how one would prove that a matrix N(x) is actually a solution without a great deal of matrix algebra. How ever, this problem is not hard as it seems. In fact, it is one of a large class of problems that can be solved via a surprising method based upon single-variable calculus. In this Journal, Khan [2] used nilpotent matrices and Taylor series to find matrix functions satisfying the exponential functional equation, f(x + y) = f(x) f(y). His method is an example of a much more general theory of matrix power series due to Weyr [4], which can be used to find matrix functions satisfying a variety of functional equations. (Rinehart [3] gives an excellent survey of Weyr's approach. Higham [1, ch. 4] gives a more comprehensive account, as well as further applications.) We say that a set of real-valued functions {f(x)}ni=x C C??(R) satisfies an analytic functional equation E if there is an analytic function E such that