On the matrix equation \(f(x)=A\) (Q1183203)

This paper deals with the matrix equation \(f(X)=A\) where \(A\in \mathbb{C}^{n\times n}\) is a given matrix and \(f\) is a complex holomorphic function defined on an open subset of \(\mathbb{C}\). In this very extensive paper the authors investigate all sorts of existence and uniqueness properties of solutions of special types for matrices \(A\) belonging to certain specified classes. Amongst others it is shown that the set of all polynomials \(P\) such that \(X=P(A)\) is a solution of \(f(X)=A\) depends only on the minimal polynomial of \(A\) and not on the size of \(A\).