Regularizing decompositions for matrix pencils and a topological classification of pairs of linear mappings

Autor(es)

Futorny, Vyacheslav ^[1] ; Rybalkina, Tetiana ^[2] ; Sergeichuk, Vladimir V. ^[2]

Afiliação do(s) autor(es)

^[1] Univ Sao Paulo, Dept Math, BR-05508 Sao Paulo - Brazil

^[2] Inst Math, Kiev - Ukraine

• Tipo de documento: Artigo Científico
• Fonte: Linear Algebra and its Applications; v. 450, p. 121-137, JUN 1 2014.
• Citações Web of Science: 2

Resumo

By Kronecker's theorem, each matrix pencil A + lambda B over a field F is strictly equivalent to its regularizing decomposition; i.e., a direct sum (I-r + lambda D) circle plus (M-1 + lambda N-1)(mt + ANt), where D is an r x r nonsingular matrix and each M-i + lambda N-i, is of the form I-k + lambda(k) (0), J(k) (0) + lambda I-k, L-k + lambda Rk, or L-k(T) + lambda R-k(T), in which L-k and R-k are obtained from I-k by deleting its last or, respectively, first row and J(k) (0) is a singular Jordan block. We give a method for constructing a regularizing decomposition of an m x n matrix pencil A + lambda B, which is formulated in terms of the linear mappings A, B : F-n -> F-m. Two m x n pencils A + lambda B and A' + lambda B' over F = R or C are said to be topologically equivalent if the pairs of linear mappings A, B : F-n -> F-m and A', B' : F-n -> F-m coincide up to homeomorphisms of the spaces F-n and F-m. We prove that two pencils are topologically equivalent if and only if their regularizing decompositions coincide up to permutation of summands and replacement of D by a nonsingular matrix D' such that the linear operators D,D' : F-r -> F-r coincide up to a homeomorphism of F-r. (C) 2014 Elsevier Inc. All rights reserved. (AU)

• Processo FAPESP: 10/50347-9 - Álgebras, representações e aplicações
• Beneficiário: Ivan Chestakov
• Modalidade de apoio: Auxílio à Pesquisa - Temático
• Processo FAPESP: 12/18139-2 - Métodos de teoria de representações em álgebra linear
• Beneficiário: Vyacheslav Futorny
• Modalidade de apoio: Auxílio à Pesquisa - Pesquisador Visitante - Internacional