Consider a $n\times n$ matrix A, say over $\mathbb{R}$ and $t \in \mathbb{R}$. In this Wikipedia article we read that
Recall from above that an $n×n$ matrix $\exp(tA)$ amounts to a linear combination of the first $n−1$ powers of $A$ by the Cayley–Hamilton theorem.
This amounts into writing $$\exp(tA) = \sum_{i=0}^{n-1}c_i(\lambda)A^n,$$ where $\lambda$ is the vector of eigenvalues of $A$ and $c(\lambda)$ functions (usually exponentials on the differences of the eigenvalues). This can be proven, as quoted above, using the Caley-Hamilton theorem and considering the characteristic polynomial of $A$.
Question: Given a sum of the form $\sum_{i=0}^{n-1}c'_i(\lambda)A^n$, can one construct some matrix exponential of the form $\exp(tf(A))$ for some function other than the identity? (the identity would correspond to $c' = c$).
Assume here that the summand is non-singular.
I would not expect this to be true in general, but I am wondering if there are some references on this precise question about going backwards through the Caley-Hamilton theorem and possibly learn what are the condition for this to hold true.
Furthermore, could we say something in the case $A$ is itself diagonal (so its entries are its eigenvalues)?