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

This article is cited in 8 scientific papers (total in 8 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

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

Full text: PDF file (121 kB)
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English version:
Functional Analysis and Its Applications, 2003, 37:3, 232–235

Bibliographic databases:

UDC: 517+512.64
Received: 01.10.2002

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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    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Bhat, BVR, “Integrators of matrices”, Linear Algebra and Its Applications, 426:1 (2007), 71  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    5. Cheung, WS, “Relationship between the zeros of two polynomials”, Linear Algebra and Its Applications, 432:1 (2010), 107  crossref  mathscinet  zmath  isi  scopus
    6. Yu. Kh. Èshkabilov, “The Efimov effect for a model “three-particle” discrete Schrödinger operator”, Theoret. and Math. Phys., 164:1 (2010), 896–904  mathnet  crossref  crossref  adsnasa  isi
    7. T. Kh. Rasulov, “On the number of eigenvalues of a matrix operator”, Siberian Math. J., 52:2 (2011), 316–328  mathnet  crossref  mathscinet  isi
    8. Kushel O., Tyaglov M., “Circulants and critical points of polynomials”, J. Math. Anal. Appl., 439:2 (2016), 634–650  crossref  mathscinet  zmath  isi  elib  scopus
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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