TITLE: Arbitrary non-integer power of a matrix
QUESTION [8 upvotes]: does there exist the notion of a non-integer power of a matrix? This seems to be accessible via semigroup-theory, yet I have not seen an actual definition so far.
I am not too firm at this right now, but I am curious. Can you give me a sketch of the definition and provide with some introductory information?

REPLY [4 votes]: There are several techniques for extending scalar functions to matrices.
Wikipedia mentions techniques based on power series, eigendecomposition, Jordan decomposition, Cauchy integral, and more.