Matrix pencils with the numerical range equal to the whole complex plane

05/10/2022

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by   Vadym Koval, et al.

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The main purpose of this article is to show that the numerical range of a linear pencil λ A + B is equal to ℂ if and only if 0 belongs to the convex hull of the joint numerical range of A and B. We also prove that if the numerical range of a linear pencil λ A + B is equal to ℂ and A + A^*, B + B^* ≥ 0, then A and B have a common isotropic vector. Moreover, we improve the classical result which describes Hermitian linear pencils.