Diagonalizing hermitian matrices of continuous functions

TL;DR

This paper investigates conditions under which hermitian matrices of continuous functions can be diagonalized, providing positive results for 2x2 matrices with differentiability, but leaving open the extension to larger matrices.

Contribution

The paper establishes diagonalization results for 2x2 hermitian matrices of continuous functions under differentiability conditions, advancing understanding in this area.

Findings

Diagonalization possible for 2x2 matrices with differentiability

Diagonalization not generally possible without additional conditions

Open problem remains for larger matrices

Abstract

The problem of diagonalizing hermitian matrices of continuous fiunctions was studied by Grove and Pederson in 1984. While diagonalization is not possible in general, in the presence of differentiability conditions we are able to obtain positive results in the case of $2\times 2$ matrices. It remains open whether our results can be extended to $n\times n$ matrices.