Zeros of Spectral Factors, the Geometry of Splitting Subspaces, and the Algebraic Riccati Inequality*

In this paper we show how the zero dynamics of (not necessarily square) spectral factors relate to the splitting subspace geometry of stationary stochastic models and to the corresponding algebraic Riccati inequality. We introduce the notion of output-induced subspace of a minimal Markovian splitting subspace, which is the stochastic analogue of the supremal output-nulling subspace in geometric control theory. Through this concept the analysis can be made coordinatefree, and straightforward geometric methods can be applied. We show how the zero structure of the family of spectral factors relates to the geometric structure of the family of minimal Markovian splitting subspaces in the sense that the relationship between the zeros of different spectral factors is reflected in the partial ordering of minimal splitting subspaces. Finally, we generalize the wellknown characterization of the the solutions of the algebraic Riccati equation in terms of Lagrangian subspaces invariant under the corresponding Hamiltonian to the larger solution set of the algebraic Riccati inequality.