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.