Abstract
We establish an analog of the Cauchy–Poincarée 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.
Similar content being viewed by others
References
N. G. de Bruijn and T. A. Springer, Indag. Math., 9, 264–270 (1947).
M. M. Malamud, Ukr. Math. J., 44, No. 12, 1658–1688 (1992).
S. M. Malamud, Linear Algebra Appl., 322, 19–41 (2001).
S. M. Malamud, Mat. Zametki, 69, Nos. 3-4, 633–637 (2001); English transl. Math. Notes, 69, No. 4, 508-514 (2001).
A. S. Markus, Usp. Mat. Nauk, 19, No. 4 (118), 93–123 (1964); English transl. Russian Math. Surveys, 19, 91-120 (1964).
M. Marcus and H. Minc, A Survey on Matrix Theory and Matrix Inequalities, Allyn and Bacon, 1964.
A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its applications, Academic Press, 1979.
G. Pólya and G. Szegö, Problems and Theorems in Analysis, Vol. II, Springer-Verlag, 1976.
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, 1988.
R. Horn and Ch. Johnson, Matrix Analysis, Cambridge, 1986.
P. Pawlowski, Trans. Amer. Math. Soc., 350, No. 11, 4461–4472 (1998).
G. Schmeisser, In: Approximation Theory: A volume dedicated to Blagovest Sendov, DARBA, Sofia, 2002, pp. 353–369.
I. Schoenberg, Amer. Math. Monthly, 93, 8–13 (1986).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Malamud, S.M. An Analog of the Poincarée Separation Theorem for Normal Matrices and the Gauss–Lucas Theorem. Functional Analysis and Its Applications 37, 232–235 (2003). https://doi.org/10.1023/A:1026044902927
Issue Date:
DOI: https://doi.org/10.1023/A:1026044902927