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

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Funktsional'nyi Analiz i ego Prilozheniya, 2003, Volume 37, Issue 3, Pages 85–88
DOI: https://doi.org/10.4213/faa162 (Mi faa162)

This article is cited in 10 scientific papers (total 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

Full-text PDF (121 kB) Citations (10)

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

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.

Received: 01.10.2002

English version:
Functional Analysis and Its Applications, 2003, Volume 37, Issue 3, Pages 232–235
DOI: https://doi.org/10.1023/A:1026044902927

Bibliographic databases:

Document Type: Article

UDC: 517+512.64

Language: Russian

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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This publication is cited in the following 10 articles:

Danielyan V S., Guterman A.E., Ng T.W., “Integrability of Diagonalizable Matrices and a Dual Schoenberg Type Inequality”, J. Math. Anal. Appl., 498:2 (2021), 124909
Walter Van Assche, “Majorization results for zeros of orthogonal polynomials”, Proc. Amer. Math. Soc., 145:9 (2017), 3849
Kushel O., Tyaglov M., “Circulants and critical points of polynomials”, J. Math. Anal. Appl., 439:2 (2016), 634–650
T. Kh. Rasulov, “On the number of eigenvalues of a matrix operator”, Siberian Math. J., 52:2 (2011), 316–328
Cheung, WS, “Relationship between the zeros of two polynomials”, Linear Algebra and Its Applications, 432:1 (2010), 107
Yu. Kh. Èshkabilov, “The Efimov effect for a model “three-particle” discrete Schrödinger operator”, Theoret. and Math. Phys., 164:1 (2010), 896–904
Bhat, BVR, “Integrators of matrices”, Linear Algebra and Its Applications, 426:1 (2007), 71
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
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
Malamud S.M., “Inverse spectral problem for normal matrices and the Gauss-Lucas theorem”, Trans. Amer. Math. Soc., 357:10 (2005), 4043–4064

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

Functional Analysis and Its Applications

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