Chain rules for functions of matrices Original Research Article

The Chain Rule that expresses the derivative of exp (A(t)) as an infinite series involving iterates of the commutator map ad A(t) is well known. We extend this formula, replacing exp with a general analytic function f, and show that its validity now depends on the location of the characteristic values of A(t) in the domain of analyticity of f. Conditions under which the Chain Rule reduces to a finite sum are also developed; some of them are closely tied to the possession by A(t) of constant generalized eigenspaces.