Simultaneous versal deformations of endomorphisms and invariant subspaces

We study the set M of pairs (f, V), defined by an endomorphism f of Fn and a d-dimensional f-invariant subspace V. It is shown that this set is a smooth manifold that defines a vector bundle on the Grassmann manifold. We apply this study to derive conditions for the Lipschitz stability of invariant subspaces and determine versal deformations of the elements of M with respect to a natural equivalence relation introduced on it.