# Arbitrary non-integer power of a matrix

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 are a few definitions for functions of matrices. See for instance http://en.wikipedia.org/wiki/Matrix_function .

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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. – J. M. Apr 18 '13 at 16:30

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 $\sqrt(A^\ast A )$? – shuhalo Jan 14 '11 at 3:46