Analytic Matrix Functions#

Definition: Analytic Matrix Function

Let \(F\) be a complete valued field and let \(f: \mathcal{D}_f \subseteq F^{n \times n} \to F^{n \times n}\) be a matrix function.

We say that \(f\) is analytic on \(S \subseteq \mathcal{D}_f\) if there exists a matrix power series \(\sum_{k \in \mathcal{I}} a_k(\boldsymbol{X} - c\boldsymbol{I}_n)^k\) which is convergent on \(S\) and is equal to \(f\):