Abstract
In this article, we introduce the weighted numerical range which is a unified approach to study the -numerical range and the rank
numerical range. If the boundaries of weighted numerical ranges of two matrices (possibly of different sizes) overlap at sufficiently many points, then the two matrices share common generalized eigenvalues.
Acknowledgements
I would like to thank C.K. Li and Karol Życzkowski for their valuable suggestions. I would also like to thank the referee for pointing out a mistake in the first version of Lemma 4.2 and other errors.
Notes
No potential conflict of interest was reported by the author.