I am currently writing a scriptum struggling with the definition of matrix power. Precisely, let $ \mathbb A \in \mathbb C^{n, n} $ and $ p \in \mathbb R $. I Currently have:
- If $ p \in \mathbb N $ then $ \mathbb A^p = \prod_{i = 1}^p \mathbb A $
- If A is hermitian and positively define and $ \mathbb A = \mathbb U^* \mathbb{DU} $ is the Schur decomposition, then $ \mathbb D $ is diagonal. We define $ \mathbb D^p $ as a diagonal matrix with $ ( \mathbb D^p )_{ii} = ( \mathbb D_{ii} )^p $. We define $ \mathbb A^p = \mathbb U^* \mathbb D^p \mathbb U $
- If $ \mathbb A = \mathbb T^{-1} \mathbb{J T} $ is the Jordan decomposition, we define $ \mathbb A^p = \mathbb T^{-1} \mathbb J^p \mathbb T $
None of these is correct however, for the first and second one have their limitations to $ p $ and $ \mathbb A $ respectively and the third one defines matrix powers using matrix powers. So, how to define matrix power so that it works with any matrix and any exponent?
I know how powers od real numbers are defined (as multiplication, later as roots for rational exponents, and finally as limits for powers of $ e $ and from that fr powers of other numbers... And then you have to prove that all of these are consistent) and I hope it won't be that complicated for matrices.
I can also prove that all of these definitions are equal (when they work).