Abstract. The possible observability indices of an observable pair of matrices, when supplementary subpairs are prescribed, are characterized when the "quotient" one is nilpotent. The geometric techniques used are also valid in the classical Carlson problem for square matrices.

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We study the set M of pairs (f, V ), defined by an endomorphism f of F n and a ddimensional 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. * Partially supported by DGICYT n.PB97-0599-C03-03, as well as DAAD D/0243869 † We are very grateful to F. Puerta for valuable comments and suggestions. ‡ alphabetical order

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