Geodesic
An Analog of the Poincar\'e Separation Theorem for Normal Matrices and the Gauss--Lucas Theorem
Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 3, pp. 85-88

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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.
Mots-clés : normal matrix
Keywords: majorization, zeros of polynomials, Gauss–Lucas theorem, Cauchy–Poincaré separation theorem, inverse problem. 
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     author = {S. M. Malamud},
     title = {An {Analog} of the {Poincar\'e} {Separation} {Theorem} for {Normal} {Matrices} and the {Gauss--Lucas} {Theorem}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {85--88},
     publisher = {mathdoc},
     volume = {37},
     number = {3},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2003_37_3_a8/}
}
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S. M. Malamud. An Analog of the Poincar\'e Separation Theorem for Normal Matrices and the Gauss--Lucas Theorem. Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 3, pp. 85-88. http://geodesic.mathdoc.fr/item/FAA_2003_37_3_a8/