My apologies if this isn't a well-enough-posed question, I think I'm partly unsure of what exact question to even ask.

There are many different ways in which we can take a function of a matrix.

- We can apply a
**holomorphic**function to any matrix, using power series, Cauchy's integral formula, or even just applying the known formula to the Jordan blocks. - We can apply any continuous (or up to measure zero, measurable) function to a
**normal**matrix by approximating with Stone-Weierstrass, or applying the function to the eigenvalues. - We can apply a
**smooth**(or even $C^k$, where $k$ is the size of the largest Jordan block) function to any**matrix with real eigenvalues**, by either using the formula on Jordan blocks or by approximating with Stone-Weierstrass in the $C^k$ norm.

My general question is therefore:

**What happens when we try to apply general smooth functions of $\mathbb{C}$ to general matrices with complex eigenvalues?**

If a matrix is diagonalizable, we can easily just diagonalize it, and apply any function to the eigenvalues. Even if it isn't diagonalizable, it seems we should still be able to apply the same formula to the Jordan blocks. While $f$ is not holomorphic, it is still totally legal to apply the differential operator $\partial_z=\frac{1}{2}(\partial_x-i\partial_y)$ to a smooth function.

Specifically, I am wondering:

- Is there something that goes
*wrong*when we try to do this? Are there discontinuities, or some other taken-for-granted property that breaks down? And if so, is this true even in the diagonalizable case? - In all the other cases of matrix functions, there are methods of computing them that don't involve actually taking a Jordan decomposition. Is there such a method for smooth functions of complex matrices? If not, is there any rigorous sense in which there
*can't be*? - Absent answers to the previous two questions, is there some other reason why I haven't ever heard of this before?