Functions of a diagonalizable matrix

Polynomial case. For $f(X)=\sum_k a_k X^k$:
$$f(A)=\sum_k a_k A^k=\sum_k a_k PD^kP^{-1}=P\left(\sum_k a_k D^k\right)P^{-1}=Pf(D)P^{-1}.$$
So the definition is consistent with the polynomial case.

Extension. For other functions (e.g. $\exp$, $\sin$, $\log$), the same formula defines $f(A)$ provided $f$ is defined on the spectrum of $A$. This requires $A$ to be diagonalizable.

Example. $e^{tA}=P\textrm{diag}(e^{t\lambda_1},\ldots,e^{t\lambda_n})P^{-1}$.

Conclusion. Functions of a diagonalizable matrix are defined via the diagonal form. For non-diagonalizable matrices, one uses Taylor or Jordan form instead.