The spectral interactor matrix for the singular Riccati equation

With reference to discrete-time linear square systems, we introduce the notion of spectral interactor matrix, which is an interactor matrix preserving the spectral properties of the underlying system. With such a matrix, a deeper insight into the properties of the solutions of the Riccati equation arising in singular filtering is gained. Precisely, we prove that the solutions can be seen as the result of two contributions, one associated with the finite zeros outside the unit circle and the second produced by the zeros at infinity (system delays)