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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
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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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http://mi.mathnet.ru/eng/faa162https://doi.org/10.4213/faa162 http://mi.mathnet.ru/eng/faa/v37/i3/p85
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This publication is cited in the following articles:
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Malamud S.M., “Inverse spectral problem for normal matrices and the Gauss-Lucas theorem”, Trans. Amer. Math. Soc., 357:10 (2005), 4043–4064
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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
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Bhat, BVR, “Integrators of matrices”, Linear Algebra and Its Applications, 426:1 (2007), 71
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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
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Cheung, WS, “Relationship between the zeros of two polynomials”, Linear Algebra and Its Applications, 432:1 (2010), 107
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Yu. Kh. Èshkabilov, “The Efimov effect for a model “three-particle” discrete Schrödinger operator”, Theoret. and Math. Phys., 164:1 (2010), 896–904
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T. Kh. Rasulov, “On the number of eigenvalues of a matrix operator”, Siberian Math. J., 52:2 (2011), 316–328
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Kushel O., Tyaglov M., “Circulants and critical points of polynomials”, J. Math. Anal. Appl., 439:2 (2016), 634–650
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