Arbitrary non-integer power of a matrix

analysis matrices

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.


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


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

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