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On the zeros of eigenpolynomials of hermitian Toeplitz matrices
Publikationstyp
Journal Article
Date Issued
2014-06-04
Author(s)
Institut
TORE-URI
Volume
80
Issue
3
Start Page
323
End Page
351
Citation
Integral Equations and Operator Theory 3 (80): 323-351 (2014)
Publisher DOI
Scopus ID
Publisher
Springer
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.
Subjects
Primary 15B05
Secondary 15B57
DDC Class
004: Informatik
510: Mathematik