Linear Algebra via Complex Analysis

Abstract

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.

Additional information

Notes on contributors

Alexander P. Campbell

ALEXANDER CAMPBELL is currently completing a Bachelor of Science at the University of Sydney. He has a longstanding interest in the connections between pure mathematics and physics. This has led to interests in many areas of mathematics, whose interactions he finds particularly inspiring. Much of his free time he spends playing piano. As an undergraduate student he enjoys performing for various theatre productions around campus.

Daniel Daners

DANIEL DANERS received his Ph.D. from the University of Zürich, Switzerland, in 1992. He currently teaches at the University of Sydney, where, given the opportunity, he likes to show students interesting mathematics outside the usual curriculum. He likes to spend his free time with his family and children.