Stratification and bundle structure of the set of conditioned invariant subspaces in the general case
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We extend some known results about the smooth stratification of the set of conditioned invariant subspaces for a general pair (C,A)€C^nXC^(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).