Let $A$ be a $k \times k$ invertible matrix over complex numbers.

If it possible to write its nth root as an analytic function (i.e. power series in $A$)?

EDIT: Complex coefficients can be functions of $A$.

## Notes

If a matrix $A$ has only one eigenvalue $\lambda$, then it is simple. We take

$$B = \exp\left[\tfrac{1}{n} \log (A ) \right]$$

where we have $B^n = A$. Using Jordan decomposition, we can simplify the logarithm to a polynomial in that matrix (as $(A - \lambda \mathbb{I})$ is nilpotent)

$$\log(A) = \log(\lambda) - \sum_{i=1}^{k} \frac{\left(- \tfrac{A}{\lambda} + \mathbb{I}_k \right)^{i}}{i}.$$