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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?

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    There is useful discussion in answers to [this duplicate question](https://math.stackexchange.com/questions/364613/can-you-raise-a-matrix-to-a-non-integer-number).2017-12-16

3 Answers 3

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You can use the binomial series to define powers for appropriate matrices.

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    Does the square root of a matrix defined with this approach coincide with the canoical square root $\sqrt(A^\ast A )$?2011-01-14
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If your matrix has positive eigenvalues, then one definition is to take non-integer powers of each eigenvalue (but keep the eigenvectors the same). This is a common definition used to take square roots, for example.

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    The more general condition is that the eigenvalues should not be negative numbers; see e.g. [this](http://books.google.com/books?hl=en&id=S6gpNn1JmbgC&pg=PA173).2013-04-18
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There are several techniques for extending scalar functions to matrices. Wikipedia mentions techniques based on power series, eigendecomposition, Jordan decomposition, Cauchy integral, and more.

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    @Mars, this is an answer from the very early days. I've tried to improve it. Thanks for the nudge.2017-12-16