On a class of functions of two non-commuting matrices: general properties, Fréchet derivatives and conditioning

By combining, multiplying, and composing primary matrix functions, one can define a class of functions of two non-commuting square matrices. This class encompasses, as special cases, well-known functions such as matrix-matrix exponentiation, the geometric mean of positive definite matrices, and the matrix beta function. It was introduced for the first time by Cardoso and Sadeghi [Numer Algor 2018;79(2):457–477]. One of the primary objectives of this work is to establish the fundamental properties of these functions and their Fréchet derivatives, thereby laying a theoretical foundation to support future research, particularly in the context of numerical computation. We present new power series representations for both the functions and their corresponding Fréchet derivatives. The numerical evaluation of the Fréchet derivative is also addressed. By extending certain results from primary matrix functions to a bivariate setting, we offer theoretical justification for using the complex step method as a reliable technique for computing both the Fréchet derivative and its associated condition number. This claim is substantiated through a series of numerical experiments.