On Determining Functions of Matrices Charles D. Allison Brigham Young University Fall 1978 Abstract This note shows how to compute analytic functions of matrix arguments. It first appeared in the Fall 1978 issue of the Pi Mu Epsilon Journal. I wondered why even though we often spoke of the matrix exponential, e At , none of my professors could tell me how to compute it. So I did some digging and came up with this. I was just finishing my first masters degree at the time. Since the time of Cayley and Sylvester there has been great interest in the computation of ma- trix functions. For example, to compute the matrix exponential e At , which satisfies the matrix differential equation with constant coefficients ˙ X (t)= AX (t), methods have been developed which rely upon properties of differential equations, the Jordan canonical form, or results from linear algebra such as normality, diagonalizability, etc.[2][4][5] Most techniques for calculating a function f of a matrix A express f (A) as a polynomial in A. Of all such methods, the simplest in concept are those based on an interpolation formula intro- duced by Sylvester[7], f (A)= n  i=1 n  j =1,j =i A - λ j I λ i - λ j f (λ i ) (1) which holds when A has distinct eigenvalues, λ 1 ,...,λ n , lying within the circle of convergence of f (z ). The notion of a matrix function is usually seen for the first time in a matrix analysis course or in a course on the theory of ordinary equations, which are graduate courses at most schools. The purpose of this note is to give a development of Sylvester’s formula accessible to the sophomore or junior in mathematics. A proof of (1) follows from the following generalization of the division algorithm, which is a modification of a theorem of Friedman[3]. Theorem 1 Let p(z ) be a polynomial with distinct roots, and let f (z ) be a function analytic in a domain D, which contains the roots of p(z ). Then there exists a unique polynomial r(z ), where deg(r)= deg(p) - 1, and a function h(z ), analytic in D, such that f (z )= p(z )h(z )+ r(z ) (2) 1