Stable noncommutative polynomials and their determinantal representations

Jurij Volčič

doi:10.1137/18M1206734

Stable noncommutative polynomials and their determinantal representations

Jurij Volčič

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

A noncommutative polynomial is stable if it is nonsingular on all tuples of matrices whose imaginary parts are positive definite. In this paper a characterization of stable polynomials is given in terms of purely stable linear matrix pencils, i.e., pencils of the form H + iP₀ + P1x1 + ··· + P_dx_d, where H is hermitian and Pj are positive semidefinite matrices. Namely, a noncommutative polynomial is stable if and only if it admits a determinantal representation with a purely stable pencil. More generally, structure certificates for noncommutative stability are given for linear matrix pencils and noncommutative rational functions.

Original language	English
Pages (from-to)	152-171
Number of pages	20
Journal	SIAM Journal on Applied Algebra and Geometry
Volume	3
Issue number	1
DOIs	https://doi.org/10.1137/18M1206734
State	Published - 1 Jan 2019
Externally published	Yes

Keywords

Determinantal representation
Hurwitz stability
Linear matrix pencil
Noncommutative polynomial
Stable polynomial

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology
Applied Mathematics

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10.1137/18M1206734

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abstract = "A noncommutative polynomial is stable if it is nonsingular on all tuples of matrices whose imaginary parts are positive definite. In this paper a characterization of stable polynomials is given in terms of purely stable linear matrix pencils, i.e., pencils of the form H + iP0 + P1x1 + ··· + Pdxd, where H is hermitian and Pj are positive semidefinite matrices. Namely, a noncommutative polynomial is stable if and only if it admits a determinantal representation with a purely stable pencil. More generally, structure certificates for noncommutative stability are given for linear matrix pencils and noncommutative rational functions.",

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N2 - A noncommutative polynomial is stable if it is nonsingular on all tuples of matrices whose imaginary parts are positive definite. In this paper a characterization of stable polynomials is given in terms of purely stable linear matrix pencils, i.e., pencils of the form H + iP0 + P1x1 + ··· + Pdxd, where H is hermitian and Pj are positive semidefinite matrices. Namely, a noncommutative polynomial is stable if and only if it admits a determinantal representation with a purely stable pencil. More generally, structure certificates for noncommutative stability are given for linear matrix pencils and noncommutative rational functions.

AB - A noncommutative polynomial is stable if it is nonsingular on all tuples of matrices whose imaginary parts are positive definite. In this paper a characterization of stable polynomials is given in terms of purely stable linear matrix pencils, i.e., pencils of the form H + iP0 + P1x1 + ··· + Pdxd, where H is hermitian and Pj are positive semidefinite matrices. Namely, a noncommutative polynomial is stable if and only if it admits a determinantal representation with a purely stable pencil. More generally, structure certificates for noncommutative stability are given for linear matrix pencils and noncommutative rational functions.

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KW - Hurwitz stability

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KW - Stable polynomial

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Stable noncommutative polynomials and their determinantal representations

Abstract

Keywords

ASJC Scopus subject areas

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