Simultaneous versal deformations of endomorphisms and invariant subspaces

We study the set of pairs (f, V), defined by an endomorphism f of 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 with respect to a natural equivalence relation introduced on it.