The prime and generalized nullspaces of right regular pencils |
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Authors: | N Karcanias G Kalogeropoulos |
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Affiliation: | (1) Control Engineering Centre, Department of Electrical, Electronic, and Information Engineering, City University, Northampton Square, EC1V 0HB London;(2) Department of Mathematics, University of Athens, Panepistimiopolis 15781, Athens, Greece |
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Abstract: | The classical notion of the -generalized nullspace, defined on a matrixA R
n×n,where is an eigenvalue, is extended to the case of ordered pairs of matrices(F, G), F, G R
m×nwhere the associated pencilsF – G is right regular. It is shown that for every C { } generalized eigenvalue of (F, G), an ascending nested sequence of spaces {P
i
,i=1, 2,...} and a descending nested sequence of spaces {ie495-02 i=1, 2,...} are defined from the -Toeplitz matrices of (F, G); the first sequence has a maximal elementM
*
, the -generalized nullspace of (F, G), which is the element of the sequence corresponding to the index ![tau](/content/qr18540853371451/xxlarge964.gif) , the -index of annihilation of (F, G), whereas the second sequence has the first elementP
*
as its maximal element, the -prime space of (F, G). The geometric properties of the {M
i
,i=1, 2,...,![tau](/content/qr18540853371451/xxlarge964.gif) and {P
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 -Toeplitz matrices of (F, G). These nested basis matrices characterize completely the geometry ofM
*
and provide a systematic procedure for the selection of maximal length linearly independent vector chains characterizing the -Segre characteristic of (F, G). |
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Keywords: | |
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