Bezoutians of Rational Matrix Functions

Various concepts of a Bezoutian of two rational matrix functions are introduced, thereby extending this concept (previously studied in the framework of matrix and operator polynomials and analystic functions) beyond the class of analytic functions. Basic properties of the Bezoutians are established. The Bezoutian based on realizations of the functions in question turns out to be most adequate and is studied in depth. In particular, its kernel is discribed in terms of the realizations. The result serves as a crucial ingredient in the proof of the key theorem of the paper, providing a characterization of the common zero data of two square-size rational matrix functions in terms of the kernel of their Bezoutian.