Unitary equivalence of normal matrices over topological spaces (Q2799114)

Given the importance of the spectral theorem in linear algebra, it is natural to ask whether it can be extended to more general situations. The generalized setting here is that the entries of the (square) matrices are complex-valued continuous functions on a topological space \(X\), with the adjoint matrix defined pointwise. This has been studied in different ways by several authors [\textit{R. V. Kadison}, Am. J. Math. 106, 1451--1468 (1984; Zbl 0585.46048); \textit{K. Grove} and \textit{G. K. Pedersen}, J. Funct. Anal. 59, 65--89 (1984; Zbl 0554.46026)]. In the present paper, the authors study the question: when are two normal matrices \(A\) and \(B\) (as introduced above) unitarily equivalent? Their approach utilizes obstruction theory from algebraic topology and yields the following result: if \(X\) is homotopically equivalent to a CW complex and \(A\) and \(B\) are multiplicity-free and have the same characteristic polynomial, then there exists a certain unique cohomology class such that \(A\) and \(B\) are unitarily equivalent if and only if this cohomology class vanishes. The authors also determine bounds on the number of unitary equivalence classes in terms of cohomological invariants of \(X\).