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Conditioning of the matrix-matrix exponentiation

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Abstract

If A has no eigenvalues on the closed negative real axis, and B is arbitrary square complex, the matrix-matrix exponentiation is defined as A B := e log(A)B. It arises, for instance, in Von Newmann’s quantum-mechanical entropy, which in turn finds applications in other areas of science and engineering. In this paper, we revisit this function and derive new related results. Particular emphasis is devoted to its Fréchet derivative and conditioning. We propose a new definition of bivariate matrix function and derive some general results on their Fréchet derivatives, which hold, not only to the matrix-matrix exponentiation but also to other known functions, such as means of two matrices, second order Fréchet derivatives and some iteration functions arising in matrix iterative methods. The numerical computation of the Fréchet derivative is discussed and an algorithm for computing the relative condition number of A Bis proposed. Some numerical experiments are included.

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Acknowledgments

We would like to thank Prof. N. J. Higham (Editor) and the anonymous reviewers for their helpful suggestions and comments. In particular, the identity (6.4) and the bound for ∥ log(A)∥ after the proof of Theorem 6.1 were suggested by one of the anonymous reviewers. The work of the first author was supported by ISR-University of Coimbra (project UID/EEA/00048/2013) funded by “Fundação para a Ciência e a Tecnologia” (FCT). The work of the corresponding author is supported by Robat Karim branch, Islamic Azad University, Tehran, Iran.

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Authors and Affiliations

  1. Polytechnic Institute of Coimbra/ISEC, Coimbra, Portugal

    João R. Cardoso

  2. Institute of Systems and Robotics, University of Coimbra, Pólo II, Coimbra, Portugal

    João R. Cardoso

  3. Department of Mathematics, Robat Karim Branch, Islamic Azad University, Tehran, Iran

    Amir Sadeghi

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  1. João R. Cardoso
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  2. Amir Sadeghi
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Correspondence to Amir Sadeghi.

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Cardoso, J.R., Sadeghi, A. Conditioning of the matrix-matrix exponentiation. Numer Algor 79, 457–477 (2018). https://doi.org/10.1007/s11075-017-0446-2

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  • DOI: https://doi.org/10.1007/s11075-017-0446-2

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