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 Frechet derivative and conditioning. We propose a new definition of bivariate matrix function and derive some general results on their Frechet derivatives, which hold, not only to the matrix-matrix exponentiation but also to other known functions, such as means of two matrices, second order Frechet derivatives and some iteration functions arising in matrix iterative methods. The numerical computation of the Frechet derivative is discussed and an algorithm for computing the relative condition number of A B is proposed. Some numerical experiments are included.
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