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
D. Alpay and H. Dym, 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 (I. Gohberg, ed.), Operator Theory: Advances and Applications OT18, Birkhäuser, Basel, 1986, pp. 89–159.
D. Alpay and H. Dym, Structured invariant spaces of vector valued rational functions, Hermitian matrices and a generalization of the Iohvidov laws, Linear Algebra Appl., in press.
D. Alpay and I. Gohberg, On orthogonal matrix polynomials, in: Orthogonal Matrix-valued Polynomials and Applications (I. Gohberg, ed.), Operator Theory: Advances and Applications, OT34, Birkhäuser, Basel, 1988, pp. 79–135.
B.D.O. Anderson and E.I. Jur, Generalized Bezoutian and Sylvester matrices in multivariable linear control, IEEE Trans. Autom. Control, AC21 (1976), 551–556.
H. Dym, Hermitian block Toeplitz matrices, orthogonal polynomials, reproducing kernel Pontryagin spaces, interpolation and extension, in: Orthogonal Matrixvalued Polynomials and Applications (I. Gohberg, ed.), Operator Theory: Advances and Applications, OT34, Birkhäuser, Basel, 1988, pp. 79–135.
H. Dym, On Hermitian block Hankel matrices, matrix polynomials, the Hamburger moment problem, interpolation and maximum entropy, Integral Equations Operator Theory, 12 (1989), 757–812.
H. Dym and N.J. Young, A Schur-Cohn theorem for matrix polynomials, Proc. Edinburgh Math. Soc., in press.
P.A. Fuhrmann, Linear Systems and Operators in Hilbert Space, McGraw Hill, New York, 1981.
M. Fujiwara, Uber die algebraischen Gleichungen, deren Wurzeln in einem Kreise oder in einer Halbebene liegen, Math. Zeit. 24 (1926), 161–169.
I. Gohberg and L. Lerer, Matrix generalizations of M. G. Krein theorems on orthogonal polynomials, in: Orthogonal Matrix-valued Polynomials and Applications (I. Gohberg, ed), Operator Theory: Advances and Applications, OT34 Birkhäuser, Basel, 1988, pp. 137–202.
G. Heinig and K. Rost, Algebraic Methods for Toeplitz-like Matrices and Operators, Operator Theory: Advances and Applications OT13, Birkhäuser, Basel, 1984.
M.G. Krein and M.A. Naimark, The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations, Linear and Multilinear Algebra, 10 (1981), 265–308.
L. Lerer and M. Tismenetsky, The Bezoutian and the eigenvalue-separation problem for matrix polynomials, Integral Equations Operator Theory, 5 (1982), 386–445.
H. Wimmer, Bezoutians of polynomial matrices, Linear Algebra Appl., 122/123/124 (1989), 475–487.
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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
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