Linear algebra via complex analysis

Abstract

The resolvent $(\lambda I-A{)}^{-1}$ of a matrix $A$ is naturally an analytic function of $\lambda \in \mathbb{C}$, and the eigenvalues are isolated singularities. We compute the Laurent expansion of the resolvent about the eigenvalues and use it to prove the Jordan decomposition theorem, the Cayley-Hamilton theorem, and to determine the minimal polynomial of $A$. The proofs do not make use of determinants and many results naturally generalise to operators on Banach spaces.