Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the conditioning of the matrix-matrix exponentiation   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
Fr\'echet derivative and conditioning. We present a general result on the
Fr\'echet derivative of bivariate matrix functions with applications not only
to the matrix-matrix exponentiation but also to other functions, such as the
second order Fr\'echet derivatives and some iteration functions arising in
matrix iterative methods. The numerical computation of the Fr\'echet derivative
is discussed and an algorithm for computing the relative condition number of
$A^B$ is proposed. Some numerical experiments are included.

18
The categories of this paper are: econ
27
The categories of this paper are: comp-gas
24
The categories of this paper are: dg-ga
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess
6
The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
22
The categories of this paper are: chao-dyn
16
The categories of this paper are: stat