# Linear algebra via complex analysis

preprint (PDF), 17 April 2012
to appear in American Mathematical Monthly

## 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.

A preprint (PDF) is available.