The classes of bivariate Schur and Herglotz matrix-valued rational functions: realizations, symmetrizations, and related determinantal representations
We present a finite-dimensional realization theory for bivariate rational functions that are contractive or have nonnegative real part on the bidisc or on the bihalfplane. We show that the realization formula depends only on the underlying domain, while the distinction between the four resulting function classes is captured entirely by explicit matrix inequalities imposed on the realization matrices. These results provide finite-dimensional realizations for rational Schur--Agler and Herglotz--Agler functions, extending the previous infinite-dimensional results. We further characterize symmetric realizations by means of a Hermitian unitary symmetry of the realization data, yielding realization theorems on the symmetrized bihalfplane. Finally, we obtain determinantal representations for symmetric stable polynomials and, consequently, for stable polynomials on the symmetrized bihalfplane. For rational functions over the real field the respective representations can use matrices with real entries.