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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

 Funktsional. Anal. i Prilozhen., 2003, Volume 37, Issue 3, Pages 85–88 (Mi faa162)

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

DOI: https://doi.org/10.4213/faa162

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

Bibliographic databases:

UDC: 517+512.64

Citation: S. M. Malamud, “An Analog of the Poincaré Separation Theorem for Normal Matrices and the Gauss–Lucas Theorem”, Funktsional. Anal. i Prilozhen., 37:3 (2003), 85–88; Funct. Anal. Appl., 37:3 (2003), 232–235

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa162
• https://doi.org/10.4213/faa162
• http://mi.mathnet.ru/eng/faa/v37/i3/p85

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Malamud S.M., “Inverse spectral problem for normal matrices and the Gauss-Lucas theorem”, Trans. Amer. Math. Soc., 357:10 (2005), 4043–4064
2. Cheung, WS, “A companion matrix approach to the study of zeros and critical points of a polynomial”, Journal of Mathematical Analysis and Applications, 319:2 (2006), 690
3. Bhat, BVR, “Integrators of matrices”, Linear Algebra and Its Applications, 426:1 (2007), 71
4. Borcea, J, “Equilibrium points of logarithmic potentials induced by positive charge distributions. I. Generalized de Bruijn-Springer relations”, Transactions of the American Mathematical Society, 359:7 (2007), 3209
5. Cheung, WS, “Relationship between the zeros of two polynomials”, Linear Algebra and Its Applications, 432:1 (2010), 107
6. Yu. Kh. Èshkabilov, “The Efimov effect for a model “three-particle” discrete Schrödinger operator”, Theoret. and Math. Phys., 164:1 (2010), 896–904
7. T. Kh. Rasulov, “On the number of eigenvalues of a matrix operator”, Siberian Math. J., 52:2 (2011), 316–328
8. Kushel O., Tyaglov M., “Circulants and critical points of polynomials”, J. Math. Anal. Appl., 439:2 (2016), 634–650
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