Matrix Power Series#

Theorem: Convergence via Real Convergence

If the real power series \(\sum_{k \in \mathcal{I}} a_k (x - c)^k\) has a convergence radius \(R \in \mathbb{R}_{\ge 0} \cup \{+\infty\}\), then the matrix power series \(\sum_{k \in \mathcal{I}} a_k (\boldsymbol{X} - c\boldsymbol{I}_n)^k\) converges for all real matrices \(\boldsymbol{X} \in \mathbb{R}^{n \times n}\) for which there exists a matrix norm of \(\boldsymbol{X} - c\boldsymbol{I}_n\) less than \(R\).