Winding number criterion for the origin to belong to the numerical range of a matrix on a loop of matrices

TL;DR

This paper establishes a winding number criterion linking the topological winding of a matrix-valued function's determinant to the inclusion of the origin in the numerical range of matrices along a loop.

Contribution

It introduces a novel topological condition involving winding numbers that determines when the origin lies in the numerical range of matrices on a loop.

Findings

If the winding number of det A is not divisible by n, the origin is in the numerical range of some matrix on the loop.

The criterion connects topological properties of matrix functions with spectral inclusion results.

Provides a new perspective on the relationship between winding numbers and numerical ranges.

Abstract

Let $A:[0,1]\to GL(n,\mathbb{C})$ be continuous with $A(0)=A(1)$, thus the winding number of $\det A$ is well-defined. If the winding number is not divisible by $n$, then the origin belongs to the numerical range of $A(\phi)$ for some $\phi \in [0,1]$.