Geometric phase and holonomy in the space of 2-by-2 symmetric operators

TL;DR

This paper introduces a novel metric on the space of 2x2 symmetric matrices, revealing its curved manifold structure and enabling efficient eigenvector computation through parallel transport, with applications in physical vibrations.

Contribution

It defines a new metric tensor on symmetric matrices that turns the space into a curved manifold, simplifying eigenvector calculations via parallel transport.

Findings

The space of symmetric matrices is shown to be a curved manifold.

Eigenvectors of matrix families can be computed by parallel transport.

The approach explains physical system vibrations through holonomy.

Abstract

We present a non-trivial metric tensor field on the space of 2-by-2 real-valued, symmetric matrices whose Levi-Civita connection renders frames of eigenvectors parallel. This results in fundamental reimagining of the space of symmetric matrices as a curved manifold (rather than a flat vector space) and reduces the computation of eigenvectors of one-parameter-families of matrices to a single computation of eigenvectors at an initial point, while the rest are obtained by the parallel transport ODE. Our work has important applications to vibrations of physical systems whose topology is directly explained by the non-trivial holonomy of the spaces of symmetric matrices.