# Linear algebra via complex analysis

preprint (PDF), 17 April 2012 (revision of 24 January 2014)
American Mathematical Monthly 120(10) (2013), 877-892
DOI: 10.4169/amer.math.monthly.120.10.877
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.