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TitleThe common invariant subspace problem: an approach via Gröbner bases
Author(s) Donu Arapura, Chris Peterson
TypeArticle in Journal
AbstractLet A be an n×n matrix. It is a relatively simple process to construct a homogeneous ideal (generated by quadrics) whose associated projective variety parametrizes the one-dimensional invariant subspaces of A. Given a finite collection of n×n matrices, one can similarly construct a homogeneous ideal (again generated by quadrics) whose associated projective variety parametrizes the one-dimensional subspaces which are invariant subspaces for every member of the collection. Gröbner basis techniques then provide a finite, rational algorithm to determine how many points are on this variety. In other words, a finite, rational algorithm is given to determine both the existence and quantity of common one-dimensional invariant subspaces to a set of matrices. This is then extended, for each d, to an algorithm to determine both the existence and quantity of common d-dimensional invariant subspaces to a set of matrices.
KeywordsEigenvector, Invariant subspace, Grassmann variety, Grobner basis, Algorithm
ISSN0024-3795
URL http://www.sciencedirect.com/science/article/pii/S0024379503008589
LanguageEnglish
JournalLinear Algebra and its Applications
Volume384
Pages1 - 7
Year2004
Edition0
Translation No
Refereed No
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