Inspired by this question, I started to wondering (firstly) if there exist any (useful / famous) matrix functions which are not defined as power series expansions with scalar coefficients, but rather matrix coefficients:
$$f({\bf A}) = \sum_{k=0}^\infty {\bf C_k A}^k, {\bf C_k,A}\in M^{n\times n}$$
I suppose that it would be more difficult to show that $\bf A$ diagonalizable $\Rightarrow f({\bf A})$ diagonalizable (in the general case) as it obviously is in the special case $\forall k: {\bf C_k}=c_k{\bf I}$ (scalar coefficients). Since identity commutes with all matrices.
(Is it even true? (secondly))