Grassmann invariants, matrix pencils, and linear system properties

AbstractThe paper deals with the establishment of relationships between two different types of invariants defined on matrix pencils and polynomial matrices and highlights their significance in the context of linear systems. For general singular matrix pencil sF - G the Grassmann invariants of the pencil are introduced as the column, row Grassmann representatives gc(F,G), gr(F,G)t, and their corresponding Plücker matrices, Pc(F,G), Pr(F,G). The properties of these new invariants of matrix pencils are established. These results provide alternative means for classifying the different families of matrix pencils, which are important for the characterization of properties in linear systems. Finally, the relationship between the Grassmann-Plücker invariants of a general rational transfer function matrix and the system matrix pencil of a minimal realization is derived. The latter results provide means for the computation of transfer function Plücker matrices from state space descriptions, as well as enable the linking of state space Kronecker invariants to the Plücker invariants of transfer functions