I was watching 3b1b
and I was wondering does there exist a definition for matrix exponentiation? Or is this concept of "matrix exponentiation" not well defined since we do not have a Taylor Series $\forall n \in\mathbb N$?
example: $2^\mathbf A$ or in general, $n^\mathbf A, \ n \in \mathbb N \land \mathbf A$ is a matrix.
I know this is probably nonsensical, but even if it isn't, it sounds interesting for how it could be used.
That makes sense, yes, as long as we are dealing with square matrices here. Define it as $\exp\bigl(\log(2)A\bigr)$, where$$\exp(M)=\operatorname{Id}+M+\frac1{2!}M^2+\frac1{3!}M^3+\cdots$$and note that this works for numbers: $\exp\bigl(\log(2)A\bigr)=\left(\exp\bigl(\log(2)\right)^A=2^A$.