Journal of Operator Theory

Volume 91, Issue 2, Spring 2024 pp. 421-441.

Chern classes and unitary equivalence of normal matrices over topological spaces

Authors: Greg Friedman (1), Efton Park (2)
Author institution: (1) Department of Mathematics, Box 298900, Texas Christian University, Fort Worth, TX 76129, U.S.A.
(2) Department of Mathematics, Box 298900, Texas Christian University, Fort Worth, TX 76129, U.S.A.

Summary: We consider unitary equivalence of normal matrices with entries in complex functions over topological spaces whose characteristic polynomials factor globally into distinct linear factors. We show such a matrix is diagonalizable if and only if the first Chern classes of its eigenbundles vanish; this implies, e.g., diagonalizability over $\mathbb{C}P^m$ for $m > 1$. Next, we show the number of equivalence classes depends only on the space and the matrix size and provide estimates for these numbers. For $\dim(X)\leqslant 3$, there are $|H^2(X)|^{n-1}$ classes of $n\times n$ matrices. Finally, for smooth manifolds, we construct de Rham cohomology classes that obstruct unitary equivalence.

DOI: http://dx.doi.org/10.7900/jot.2022may27.2384
Keywords: normal matrices, unitary equivalence, obstruction theory, characteristic classes, eigenvalues, eigenvectors

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