On the geometry of the generalised nullspace of right regular pencils

The classical notion of the λ-generalized nullspace, defined on a matrix A∈R^n×n, where λ is an eigenvalue, is extended to the case of ordered pairs of matrices (F,G),F,G∈R ^m×n, where the associated pencil sF-G is right regular. It is shown that, for every α∈C∪ {∞}, generalized eigenvalue of (F ,G), an ascending nested sequence of spaces {M ⁱ_α, i=1, 2, . . .,} is defined from the α-Toeplitz matrices of (F;G); this sequence has a maximal element M*_α, the α-generalized null space of (F,G), which is the element of the sequence corresponding to the index τ_α , the α-index of annihilation of (F,G). The geometric properties of the {Mⁱ_α, i=1, 2, . . ., τ _α} set are investigated and are shown to be intimately related to the existence of nested basis matrices of the null spaces of the α-Toeplitz matrices of (F,G); these nested basis matrices characterize completely the geometry of M*_α and provide a systematic procedure for the selection of maximal length linearly independent vector chains characterizing the α-Segre´ characteristic of (F,G )