This article is cited in 10 papers

Brief communications

An Analog of the Poincaré Separation Theorem for Normal Matrices and the Gauss–Lucas Theorem

S. M. Malamud

Swiss Federal Institute of Technology

Abstract: We establish an analog of the Cauchy–Poincaré 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.

Keywords: normal matrix, majorization, zeros of polynomials, Gauss–Lucas theorem, Cauchy–Poincaré separation theorem, inverse problem.

UDC: 517+512.64

Received: 01.10.2002

DOI: 10.4213/faa162

 English version: Functional Analysis and Its Applications, 2003, 37:3, 232–235

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