On the geometry of the generalised nullspace of right regular pencils
N. Karcanias, G. Kalogeropoulos · 2002
The classical notion of the lambda -generalized nullspace, defined on a matrix A in R/sup n*n/, where lambda is an eigenvalue, is extended to the case of ordered pairs of matrices (F,G),F,G in R/sup m*n/, where the associated pencil sF-G is right regular. It is shown that, for every alpha in C union ( infinity ), generalized eigenvalue of (F,G), an ascending nested sequence of spaces (M/sup i//sub alpha /, i=1, 2, . . .,) is defined from the alpha -Toeplitz matrices of (F;G); this sequence has a maximal element M*/sub alpha /, the alpha -generalized null space of (F,G), which is the element of the sequence corresponding to the index tau /sub alpha /, the alpha -index of annihilation of (F,G). The geometric properties of the (M/sup i//sub alpha /, i=1, 2, . . ., tau /sub alpha /) set are investigated and are shown to be intimately related to the existence of nested basis matrices of the null spaces of the alpha -Toeplitz matrices of (F,G); these nested basis matrices characterize completely the geometry of M*/sub alpha / and provide a systematic procedure for the selection of maximal length linearly independent vector chains characterizing the alpha -Segre' characteristic of (F,G).>