On the Fréchet derivative of matrix functions

Abstract In 1956 Rinehart [4] discussed the derivatives of matrix functions by considering differences ƒ( A + E ) − ƒ( A ) for matrices E commuting with A . In that case the derivative turned out to be ƒ′ (A) . In this paper the case of noncommutative A and E is treated. This leads to the Frechet derivative of the matrix function ƒ. An explicit integral representation is obtained. Using an approach that is similar to the one in [5], a finite sum in polynomials of A is obtained. The coefficients may be computed recursively. This is useful in computing the Frechet derivative which is needed for Newton's method to solve nonlinear matrix equations.