Higher order derivatives of matrix functions

We present theory for general partial derivatives of matrix functions on the form $f(A(x))$ where $A(x)$ is a matrix path of several variables ($x=(x_1,\dots ,x_j)$). Building on results by Mathias [SIAM J. Matrix Anal. Appl., 17 (1996), pp. 610-620] for the first order derivative, we develop a block upper triangular form for higher order partial derivatives. This block form is used to derive conditions for existence and a generalized Daleckiĭ-Kreĭn formula for higher order derivatives. We show that certain specializations of this formula lead to classical formulas of quantum perturbation theory. We show how our results are related to earlier results for higher order Fréchet derivatives. Block forms of complex step approximations are introduced and we show how those are related to evaluation of derivatives through the upper triangular form. These relations are illustrated with numerical examples.