A Daleckii-Krein formula for the Frechet derivative of a generalized matrix function

Noferini, Vanni (2016) A Daleckii-Krein formula for the Frechet derivative of a generalized matrix function. [MIMS Preprint]

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Abstract

We state and prove an extension of the Daleckii-Krein theorem, thus obtaining an explicit formula for the Frechet 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 generalized matrix functions are real differentiable but generally not complex differentiable. Finally, we discuss the application of our result 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.

Item Type: MIMS Preprint
Uncontrolled Keywords: generalized matrix function, Daleckii-Krein theorem, Gateaux derivative, Frechet derivative, condition number
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Dr V Noferini
Date Deposited: 10 May 2016
Last Modified: 08 Nov 2017 18:18
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2466

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