  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 strongly 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 strongly stable pencil. More generally, structure
certificates for noncommutative stability are given for linear matrix pencils
and noncommutative rational functions.
