Chern classes and unitary equivalence of normal matrices over topological spaces

TL;DR

This paper investigates the conditions under which normal matrices over topological spaces are unitarily equivalent, linking diagonalizability to Chern classes and classifying equivalence classes via cohomology.

Contribution

It establishes criteria for diagonalizability based on Chern classes, classifies unitary equivalence classes over CW complexes, and introduces cohomological obstructions for smooth matrices.

Findings

Diagonalizability linked to vanishing first Chern classes.

Classification of unitary equivalence classes over CW complexes.

Cohomology classes as obstructions in smooth cases.

Abstract

This paper continues the authors' work on the question of unitary equivalence of matrices with entries in the complex-valued functions of a topological space (matrices over spaces). Specifically, we here consider the question of unitary equivalence for pairs of normal matrices over a space that share a common characteristic polynomial that can be globally factored into distinct linear factors. We show that such a matrix is diagonalizable if and only if the first Chern classes of its eigenbundles all vanish and derive as an application that all such matrices over $\mathbb{C}P^m$ are diagonalizable for $m > 1$. Next, given a CW complex $X$ and a polynomial $\mu$ in $C(X)[\lambda]$ that globally splits into distinct linear factors, we prove that the number of unitary equivalence classes of matrices with $\mu$ as a characteristic polynomial depends only on the space $X$ and the degree of $\mu$, and we give some estimates on how many unitary equivalence classes there can be. In the case that $X$ is a CW complex of dimension at most three, we demonstrate a bijection between the unitary equivalence classes of $n \times n$ normal matrices with characteristic polynomial $\mu$ and elements of the group $(H^2(X))^{n-1}$. Finally, when $X$ is a smooth manifold and we restrict to matrices with smooth entries, we construct a de Rham cohomology class whose nonvanishing is an obstruction to unitary equivalence.