Speaker: Greg Friedman, Texas Christian University

Title: Topological linear algebra - the unitary equivalence of matrices over topological spaces

(joint with Efton Park, Texas Christian University)

Abstract: As an application of topology to linear algebra, we study the question of unitary equivalence for matrices with coefficients in the ring C(X) of continuous complex-valued functions over a topological space X (equivalently, one can imagine a family of matrices that varies continuously over the points of X). If X is a CW complex and A and B are normal matrices over C(X) with the same set of distinct eigenvalues, we show that there is a single cohomological obstruction that completely determines whether A and B are unitary equivalent. Using this tool, it is often possible to count or bound the number of unitary equivalence classes on X with given eigenvalues. This generalizes work of Grove and Pedersen on the diagonalizability of normal matrices with coefficients in C(X).