ISSN:
1531-5878
Source:
Springer Online Journal Archives 1860-2000
Topics:
Electrical Engineering, Measurement and Control Technology
Notes:
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 α i ,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 τα, 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 α i ,i=1, 2,...,τα and {P α i ,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).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01260334
Permalink
