On the conditioning of the matrix-matrix exponentiation
This work provides theoretical and computational tools for analyzing the sensitivity of a bivariate matrix function relevant to quantum entropy and other applications, but the results are incremental for the matrix function community.
The paper derives new results on the Fréchet derivative and conditioning of the matrix-matrix exponentiation A^B, proposing an algorithm to compute its relative condition number. Numerical experiments demonstrate the algorithm's effectiveness.
If ${A}$ has no eigenvalues on the closed negative real axis, and $B$ is arbitrary square complex, the matrix-matrix exponentiation is defined as $A^B:=e^{\log({A}){B}}$. This function arises, for instance, in Von Newmann's quantum-mechanical entropy, which in turn finds applications in other areas of science and engineering. Since in general $A$ and $B$ do not commute, this bivariate matrix function may not be a primary matrix function as commonly defined, which raises many challenging issues. In this paper, we revisit this function and derive new related results. Particular emphasis is given to its Fréchet derivative and conditioning. We present a general result on the Fréchet derivative of bivariate matrix functions with applications not only to the matrix-matrix exponentiation but also to other functions, such as the 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.