How to define complex number power of matrix?

Question

Rational number or real number power of some $n*n$ matrix is able to be defined, can we define the complex number power of matrix like $S_{n\times n}^i $ $i$ is the complex number.

Answer 1

I believe the way to calculate this would be to utilize the natural logarithm of the matrix S;

$$S^i=e^{i\ln(S)}=\sum_{n=0}^{\infty}{i^n\over n!}\cdot ln^n(S)$$

$$=\sum_{n=0}^{\infty}{i^n \cdot(-1)^n\over n!}\cdot\left(\sum_{k=1}^{\infty}{(I-S)^k\over k}\right)^n$$