Stable noncommutative polynomials and their determinantal\n representations

2018/07/15 by Jurij Volčič, Volčič, Jurij
Computer Science · Mathematics · #Matrix Theory and Algorithms #Advanced Topics in Algebra #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.1807.05645

Abstract

A noncommutative polynomial is stable if it is nonsingular on all tuples of\nmatrices whose imaginary parts are positive definite. In this paper a\ncharacterization of stable polynomials is given in terms of strongly stable\nlinear matrix pencils, i.e., pencils of the form H+iP₀+P₁x₁+\⋯+P_dx_d,\nwhere H is hermitian and P_j are positive semidefinite matrices. Namely, a\nnoncommutative polynomial is stable if and only if it admits a determinantal\nrepresentation with a strongly stable pencil. More generally, structure\ncertificates for noncommutative stability are given for linear matrix pencils\nand noncommutative rational functions.\n

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