A Formula for the Fréchet Derivative of a Generalized Matrix Function

Abstract

We state and prove an analogue of the Daleckii--Krein theorem, thus obtaining an explicit formula for the Fréchet derivative of generalized matrix functions. Moreover, we prove the differentiability of generalized matrix functions of real matrices under very mild assumptions. For complex matrices, we argue that, under the same assumptions, generalized matrix functions are real-differentiable but generally not complex-differentiable. Finally, we discuss the application of our results to the study of the condition number of generalized matrix functions. Along our way, we also derive generalized matrix functional analogues of a few classical theorems on polynomial interpolation of classical matrix functions and their derivatives.