Is the operation taking a matrix to the power of another matrix well-defined?

Question

e.g. if A and B are matrices, is there a useful definition for $A^B$? I don't see an obvious definition; but then the definition of the matrix exponential also would never occur to me independently, but it makes perfect sense. Unfortunately I can't see an obvious way to extend that sensible definition into the general operation I'm looking for, since the matrix logarithm is multivalued. Is there a way around this?

Answer 1

If you assume that $A$ is normal and nonsingular (and perhaps a bit more), then you can define $A^B$ by matrix-matrix eponentiation as in http://www.rockefeller.edu/labheads/cohenje/PDFs/215BarrabasCohenalApp19941.pdf, section $3$. Of course, it uses $\exp$ and $\log$ for matrices. To be more precise, $A^B=e^{(\log A)B}$, or $A^B=e^{B(\log A)}$. Even exponential product formulas are derived there: $$ A^{B+C}=\lim_{k\to \infty}(A^{B/k}A^{C/k})^k, $$ relying on Lie's exponential product formula.