Stratification and bundle structure of the set of conditioned invariant subspaces in the general case

Abstract

We extend some known results about the smooth stratification of the set of conditioned invariant subspaces for a general pair $(C,A)\in\w C^n\times\w C^{n+m}$ without any assumption on the observability. More precisely we prove that the set of $(C,A)$-conditioned invariant subspaces having a fixed Brunovsky-Kronecker structure is a submanifold of the corresponding Grassman manifold, with a vector bundle structure relating the observable and nonobservable part, and we compute its dimension. We also prove that the set of all $(C,A)$-conditioned invariant subspaces having a fixed dimension is connected, provided that the nonobservable part of $(C,A)$ has at most one eigenvalue (this condition is in general necessary).