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Integral Equations and Operator Theory

, Volume 80, Issue 3, pp 323–351 | Cite as

On the Zeros of Eigenpolynomials of Hermitian Toeplitz Matrices

  • Florian BüngerEmail author
Article
  • 102 Downloads

Abstract

This article refines a result of Delsarte, Genin, Kamp (Circuits Syst Signal Process 3:207–223, 1984), and Delsarte and Genin (Springer Lect Notes Control Inf Sci 58:194–213, 1984), regarding the number of zeros on the unit circle of eigenpolynomials of complex Hermitian Toeplitz matrices and generalized Caratheodory representations of such matrices. This is achieved by exploring a key observation of Schur (Über einen Satz von C. Carathéodory. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften, pp. 4–15, 1912) stated in his proof of a famous theorem of Carathéodory (Rendiconti del Circolo Matematico di Palermo 32:193–217, 1911). In short, Schur observed that companion matrices corresponding to eigenpolynomials of Hermitian Toeplitz matrices H define isometries with respect to (spectrum shifted) submatrices of H. Looking at possible normal forms of these isometries leads directly to the results. This geometric, conceptual approach can be generalized to Hermitian or symmetric Toeplitz matrices over arbitrary fields. Furthermore, as a byproduct, Iohvidov’s law in the jumps of the ranks and the connection between the Iohvidov parameter and the Witt index are established for such Toeplitz matrices.

Mathematics Subject Classification (2010)

Primary 15B05 Secondary 15B57 

Keywords

Hermitian Toeplitz matrices zeros of eigenpolynomials Iohvidov parameter 

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Institute for Reliable ComputingHamburg University of TechnologyHamburgGermany

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