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Winding number criterion for the origin to belong to the numerical range of a matrix on a loop of matrices
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classification
math.FAmath-phmath.MPmath.OAmath.SPphysics.optics
keywords
numberwindingnumericaloriginrangebelongbelongscontinuous
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 $\phi \in [0,1]$.
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