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Considering $A^b$, we have a nice definition and properties when $A$ is a matrix and $b$ is a real (or field) value.

We also have a fairly useful definition for $e^B$, where $B$ is a matrix and $e$ is the Naperian base. I imagine this can be extended to an arbitrary base using logarithms.

My question is whether there is a general definition for $A^B$ with $A$ and $B$ as matrices with some suitable properties. Some properties might be:

  • $A^b$ and $e^B$ are special cases of the definition of $A^B$
  • $A^I = A$
  • $A^{B + C} = A^BA^C \text{ under suitable conditions }$
  • $(A^B)^C = A^{BC} \text{ under suitable conditions }$

Is any such definition known? Can computations be done with it? What properties do we get and under what conditions? Thank you, this is for my own curiosity.

Edit : Fly by Night found this previous question: Matrix raised to a matrix, but it doesn't really address the properties that I was asking about. The real usefulness in a definition is in what you can infer from it, so I want to keep this question open.

Community
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DanielV
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  • you may like to see the new answer here: https://math.stackexchange.com/questions/164422/matrix-raised-to-a-matrix/2527450#2527450 – Guy Fsone Nov 19 '17 at 12:41

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If $A$ is "nice enough" (being diagonalizable with strictly positive eigenvalues should do the job), you can define the logarithm of $A$ by the power series: $$\log A = \log(1+(A-1)) = \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}(A-1)^n$$ Then you have $A=\exp(\log A)$ and thus: $$A^B=\exp(B\log A)$$ (under suitable conditions).

Daniel Robert-Nicoud
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