Unitary equivalence of normal matrices over topological spaces

TL;DR

This paper characterizes when two normal matrices with continuous function entries over a topological space are unitarily equivalent, using obstruction theory and cohomological invariants, providing a complete criterion and bounds on equivalence classes.

Contribution

It introduces a necessary and sufficient condition for unitary equivalence of such matrices and establishes bounds based on topological invariants, advancing the understanding of matrix equivalence over topological spaces.

Findings

Obstruction theory provides a criterion for unitary equivalence.

Bounds on the number of equivalence classes are derived.

Cohomological invariants determine the classification complexity.

Abstract

Let A and B be normal matrices with coefficients that are continuous complex-valued functions on a topological space X that has the homotopy type of a CW complex, and suppose these matrices have the same distinct eigenvalues at each point of X. We use obstruction theory to establish a necessary and sufficient condition for A and B to be unitarily equivalent. We also determine bounds on the number of possible unitary equivalence classes in terms of cohomological invariants of X.