The famous Cayley-Hamilton theorem states that a matrix $\bf A$ with eigenvalues $$\{\lambda_k({\bf A})\} \hspace{0.5cm} \text{ s.t. } \hspace{0.5cm} \det({\bf A}-\lambda_k({\bf A}){\bf I}) =0, \forall k$$ must be a root to it's own characteristic polynomial:
$$p(x) = \prod_{\forall_i}^n(x-\lambda_i({\bf A})) = c_0+c_1x+\cdots x^n$$
so that
$$p({\bf A}) = c_0{\bf I}+c_1{\bf A} + \cdots +{\bf A}^n = 0$$
Now all $c_k$ are scalars so in practice each monomial in $\bf A$ is multiplied by a diagonal matrix and then summed together, the matrix is forced to be a scalar linear combination of it's own higher powers and the identity.
Does there exist any generalization : any other polynomial that $\bf A$ needs to fulfill where $c_k$ could be matrices which are (possibly) tied to other properties of the matrix than the scalar eigenvalues $\lambda_k$?
I believe this is what you're looking for:
A generalized Cayley-Hamilton theorem for polynomials with Matrix Coefficients -Hwang