The general solution of the equation \(f(\lambda +1)=\lambda f(\lambda)\) is \(f(\lambda)=h(\lambda)\Gamma (\lambda),\) where \(\Gamma\) is the gamma function and h is an arbitrary function of period 1. The principal result in this paper is the determination of the matrix-valued functions f of the matrix variable A which satisfy the functional equations (1) \(f(A+1)=Af(A)=f(A)A\). The first half of the paper is a survey giving the needed background. In particular, it discusses Giorgi's definition of a matrix-valued function associated with a (sufficiently differentiable) scalar valued function f. If A has the canonical form \(P^{-1}AP=J_ 1\oplus J_ 2\oplus...\oplus J_ k\), then f(A) is given by \((2)\quad f(A)=P(f(J_ 1)\oplus...\oplus f(J_ k))P^{-1};\) if the Jordan block \(J_ i=\lambda_ iI+B_ i\) has size m(i), then \((3)\quad f(J_ i)=\sum (1/j!)f^{(j)}(\lambda_ i)B^ j_ i,\) the sum taken for \(0\leq j\leq m(i)-1\). The authors give an extension of Giorgi's definition; instead of a single function f and its derivatives, they have a finite sequence of functions \((f_ 0,...,f_{n-1})\) and \(f_ j(\lambda_ i)\) replaces \(f^{(j)}(\lambda_ i)\) in (3). With this machinery in place, the authors solve (1) for the case where \(A=J=\lambda I+B\) is a Jordan block of size m. This is equivalent to the solution of the system of difference equations \(f_ 0(\lambda +1)-\lambda f_ 0(\lambda)=0,\) \(f_ j(\lambda +1)-\lambda f_ j(\lambda)=f_{j-1}(\lambda),\) \(j=1,...,m-1\), and then \(f(J)=\sum f_ j(\lambda)B^ j,\) the sum taken for \(0\leq j\leq m(i)-1\). Finally, the solution of (1) for general A is obtained using (2).