Linear algebra via complex analysis

The resolvent (λI − A)⁻¹ of a matrix A is naturally an analytic function of λ ∈ ℂ, and the eigenvalues are isolated singularities. We compute the Laurent expansion of the resolvent about the eigenvalues of A. Using the Laurent expansion, we prove the Jordan decomposition theorem, prove the Cayley–Hamilton theorem, and determine the minimal polynomial of A. The proofs do not make use of determinants, and many results naturally generalize to operators on Banach spaces.