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

2023/11/01 by Cheng Guo, Shanhui Fan, Guo, Cheng +1
Computer Science · Mathematics · Engineering · #Matrix Theory and Algorithms #Advanced Topics in Algebra #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2311.00849

Abstract

Let A:[0,1]→ GL(n,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(ϕ) for some ϕ∈ [0,1].

Related

• The winding number of a closed curve around a point
• Winding numbers and Fourier series
• One cannot hear the winding number
• A discrete formulation for three-dimensional winding number
• The Winding Number at the Critical Hölder Exponent 1/3: Failure of Universal Fourier Summation