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Topics in Matrix and Operator Theory pp 191–214Cite as

Birkhäuser

A Hermite Theorem for Matrix Polynomials

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 50))

Abstract

An analogue of the Hermite theorem for the number of zeros in a half plane for a scalar polynomial is obtained for a class of m × m matrix polynomials by (finite dimensional) reproducing kernel Krein space methods. The paper, which is largely expository, is partially modelled on an earlier paper with N.J. Young which developed similar analogues of the Schur-Cohn theorem for matrix polynomials. More complete results in a somewhat different formulation have been obtained by Lerer and Tismenetsky by other methods. New proofs of some recent results on the distribution of the roots of certain matrix polynomials which are associated with invertible Hermitian block Hankel and block Toeplitz matrices are presented as an application of the main theorem.

The author wishes to express his thanks to Renee and Jay Weiss for endowing the chair which supported this research.

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References

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Author information

Authors and Affiliations

  1. Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel

    Harry Dym

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  1. Harry Dym
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Editor information

Editors and Affiliations

  1. Econometric Institute, Erasmus University Rotterdam, Postbus 1738, 3000 DR, Rotterdam, The Netherlands

    H. Bart , I. Gohberg  & M. A. Kaashoek ,  & 

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© 1991 Springer Basel AG

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Dym, H. (1991). A Hermite Theorem for Matrix Polynomials. In: Bart, H., Gohberg, I., Kaashoek, M.A. (eds) Topics in Matrix and Operator Theory. Operator Theory: Advances and Applications, vol 50. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5672-0_8

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  • DOI: https://doi.org/10.1007/978-3-0348-5672-0_8

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-5674-4

  • Online ISBN: 978-3-0348-5672-0

  • eBook Packages: Springer Book Archive

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