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

The numerical range of a matrix is a compact convex set in the complex plane that contains the convex hull of the spectrum of the matrix. A key question is whether the origin belongs to the convex hull of the spectrum or the numerical range. This question arises in many computational and engineering applications such as the stability analysis of dynamic systems and the study of scattering matrices. Yet, there lacks an analytical criterion for this problem. Many relevant applications involve parameter-dependent matrices. In this paper, we provide a simple sufficient analytical criterion for the zero-inclusion problem, for a loop of parameter-dependent matrices. We prove the following theorem: Let [Formula: see text] (the set of [Formula: see text] invertible complex matrices) be continuous with [Formula: see text], thus the winding number of [Formula: see text] is well defined. If the winding number is not divisible by n, then the origin belongs to the convex hull of the spectrum, thus the numerical range of [Formula: see text] for some [Formula: see text]. Our criterion may find potential applications in physics, engineering, and computation.