We are given $f(\lambda)$ any suitable function in our context here and $n\times n$ matrix $A$ with characteristic polynomial $p(\lambda)=\Pi_{i=1}^{m} (\lambda -\lambda_i)^{n_i}$ where $n=\sum_{i=1}^{m}n_i$, we define $h(\lambda)=\beta_0+\dots+b_{n-1}\lambda^{n-1}$, unknown $\beta_i$'s are to be found by followings:
$f^{(k)}(\lambda_i)=h^{(k)}(\lambda_i)\text{ for } k=0,1,\dots, n_i-1\text{ and } i=1,2,\dots, m$ where $f^{(k)}(\lambda_i)= \frac{d^k f(\lambda)}{d\lambda^k}|_{\lambda=\lambda_i}$
and $h^{(k)}(\lambda_i)$ is similarly defined. then we have $f(A)=h(A)$ ,and $f(\lambda)=h(\lambda)$ on the spectrum of $A$.
My question is this process is valid or will work even if the algebraic multiplicity of some eigenvalue is not equal to its geometric multiplicity Since The statements/method don't say anything about am and gm of an eigenvalue.