Stable noncommutative polynomials and their determinantal representations

Jurij Volčič

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


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.

Original languageEnglish
Pages (from-to)152-171
Number of pages20
JournalSIAM Journal on Applied Algebra and Geometry
Issue number1
StatePublished - 1 Jan 2019
Externally publishedYes


  • 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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