Stable Noncommutative Polynomials and Their Determinantal Representations
Jurij Volčič · SIAM Journal on Applied Algebra and Geometry · 2019
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 $H+iP_0+P_1x_1+\cdots+P_dx_d$, where $H$ is hermitian and $P_j$ 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.