Colloquium: "Unitary Equivalence of Normal Matrices over Topological Spaces"

Host: John McCarthy
Tea @ 3:45pm in Cupples I, Room 200

Abstract: One of the most striking theorems in linear algebra is the spectral theorem: every normal matrix with complex entries is diagonalizable. An immediate consequence of the spectral theorem is that a normal matrix over the complex numbers is determined up to unitary equivalence by its eigenvalues, counting multiplicities. Suppose we have a normal matrix with entries that are continuous functions on some topological space. Under what circumstances does the spectral theorem still hold? Given two such matrices, when are they unitarily equivalent? These questions and related ones have interesting and nontrivial answers, and it turns out that algebraic topology is a useful tool for studying such questions, as I will (I hope!) demonstrate. Anyone who has taken a linear algebra course and a first course in algebraic topology should be well-equipped to understand this talk.