Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-18T03:43:27.595Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

REFERENCES

1.Alpay, D. and Dym, H., On applications of reproducing kernel spaces to the Schur algorithm and rational J unitary factorization, in: I. Schur Methods in Operator Theory and Signal Processing (Operator Theory: Advances and Applications, OT 18, Birkhäuser-Verlag, Basel, 1986), 89–159.CrossRefGoogle Scholar
2.Anderson, B. D. O. and Jury, E. I., Generalized Bezoutian and Sylvester matrices in multivariable linear control, IEEE Trans. Automat. Control 21 (1976), 551–556.CrossRefGoogle Scholar
3.Ball, J. A., Gohberg, I. C. and Rodman, L., Realization and interpolation of rational matrix functions, in: Topics in Interpolation Theory of Rational Matrix-valued Functions (Operator Theory: Advances and Applications, OT 33, Birkhäuser-Verlag, Basel, 1988), 1–72.Google Scholar
4.Cohn, A., Über die Anzahl der Wurzeln einer algebraischen Gleichung in Einem Kreise, Math. Z. 14 (1922), 110–148.CrossRefGoogle Scholar
5.Dewilde, P., Input-output description of roomy systems, SIAM J. Control Optim. 14 (1976), 712–736.CrossRefGoogle Scholar
6.Fuhrmann, P. A., On the corona theorem and its applications to spectral problems in Hilbert space, Trans. Amer. Math. Soc. 132 (1968), 55–66.Google Scholar
7.Gohberg, I., Lancaster, P. and Rodman, L., Matrix Polynomials (Academic Press, New York, 1982).Google Scholar
8.Kailath, T., Linear Systems (Prentice Hall, Englewood Cliffs, New Jersey, 1980).Google Scholar
9.Lerer, L. and Tismenetsky, M., The Bezoutian and the eigenvalue-separation problem for matrix polynomials, Integral Equations Operator Theory 5 (1982), 386–445.CrossRefGoogle Scholar
10.MacDuffee, C. C., The Theory of Matrices (Springer, Berlin 1933; reprinted by Chelsea Publishing Co., New York, 1950).Google Scholar
11.Schur, I., Über potenzreihen, die im Innern des Einheitskreises beschränkt sind, J. für Mathematik 147 (1917), 205–232 and 148 (1918), 122–145; English Transl. in: J. Schur Methods in Operator Theory and Signal Processing (Operator Theory: Advances and Applications, OT 18, Birkhäuser-Verlag, Basel, 1986), 31–88.Google Scholar
12.Dym, H., Hermitian block Toeplitz matrices, orthogonal polynomials, reproducing kernel Pontryagin spaces, interpolation and extension, in: Orthogonal Matrix-valued Polynomials and Applications (Operator Theory: Advances and Applications, OT 34, Birkhäuser-Verlag, Basel, 1988), 79–135.CrossRefGoogle Scholar