The prime and generalized nullspaces of right regular pencils

Abstract

The classical notion of the λ-generalized nullspace, defined on a matrixA εR ^n×n,where λ is an eigenvalue, is extended to the case of ordered pairs of matrices(F, G), F, G ε R ^m×nwhere the associated pencilsF − G is right regular. It is shown that for every α εC ∪ {∞} generalized eigenvalue of (F, G), an ascending nested sequence of spaces {P ⁱ_α ,i=1, 2,...} and a descending nested sequence of spaces {ie495-02 i=1, 2,...} are defined from the α-Toeplitz matrices of (F, G); the first sequence has a maximal elementM ^*_α , the α-generalized nullspace of (F, G), which is the element of the sequence corresponding to the index τ_α, the α-index of annihilation of (F, G), whereas the second sequence has the first elementP ^*_α as its maximal element, the α-prime space of (F, G). The geometric properties of the {M ⁱ_α ,i=1, 2,...,τ_α and {P ⁱ_α ,i=1, 2,...sets, as well as their interrelations are investigated and are shown to be intimately related to the existence of nested basis matrices of the nullspaces of the α-Toeplitz matrices of (F, G). These nested basis matrices characterize completely the geometry ofM ^*_α and provide a systematic procedure for the selection of maximal length linearly independent vector chains characterizing theα-Segre characteristic of (F, G).