Abstract
We establish an analog of the Cauchy–Poincarée separation theorem for normal matrices in terms of majorization. A solution to the inverse spectral problem (Borg type result) is also presented. Using this result, we generalize and extend the Gauss–Lucas theorem about the location of roots of a complex polynomial and of its derivative. The generalization is applied to prove old conjectures due to de Bruijn–Springer and Schoenberg.
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Malamud, S.M. An Analog of the Poincarée Separation Theorem for Normal Matrices and the Gauss–Lucas Theorem. Functional Analysis and Its Applications 37, 232–235 (2003). https://doi.org/10.1023/A:1026044902927
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DOI: https://doi.org/10.1023/A:1026044902927