Conditioning of the matrix-matrix exponentiation

Original Paper


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 B is proposed. Some numerical experiments are included.


Matrix-matrix exponentiation Bivariate matrix function Conditioning Fréchet derivative Matrix exponential Matrix logarithm 


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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. 1.Polytechnic Institute of Coimbra/ISECCoimbraPortugal
  2. 2.Institute of Systems and RoboticsUniversity of Coimbra, Pólo IICoimbraPortugal
  3. 3.Department of Mathematics, Robat Karim BranchIslamic Azad UniversityTehranIran

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