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A Schur-Cohn theorem for matrix polynomials

Published online by Cambridge University Press:  20 January 2009

Harry Dym  and
Nicholas Young
Harry Dym
Affiliation:
Department of Theoretical MathematicsThe Weizmann Institute of ScienceRehovot 76100, Israel
Nicholas Young
Affiliation:
Department of MathematicsUniversity of LancasterLancaster LA1 4YF, U.K.
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Abstract

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Let N(λ) be a square matrix polynomial, and suppose det N is a polynomial of degree d. Subject to a certain non-singularity condition we construct a d by d Hermitian matrix whose signature determines the numbers of zeros of N inside and outside the unit circle. The result generalises a well known theorem of Schur and Cohn for scalar polynomials. The Hermitian “test matrix” is obtained as the inverse of the Gram matrix of a natural basis in a certain Krein space of rational vector functions associated with N. More complete results in a somewhat different formulation have been obtained by Lerer and Tismenetsky by other methods.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 33 , Issue 3 , October 1990 , pp. 337 - 366
Copyright
Copyright © Edinburgh Mathematical Society 1990

References

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