Matrix continued fractions are directly related to the maximal (A, B)-invariant subspace in Ker

The maximal -invariant subspace in Ker is characterized in terms of polynomial models, by its properties pertaining to the Hankel map. This characterization is directly related to the first stage in a continued fraction representation of the transfer function. The atoms of the continued fraction are given explicitly in terms of Tocplitz operators. We also obtain a complete characterization and parametrization of eligible first-atoms, based on Weiner-Hopf indexes. We show how to construct a first atom which is column proper and determines a feedback equivalent, feedback irreducible transfer function.