Specht's criterion for systems of linear mappings

Autor(es)

Futorny, Vyacheslav ; Horn, Roger A. ; Sergeichuk, Vladimir V.

• Tipo de documento: Artigo Científico
• Fonte: Linear Algebra and its Applications; v. 519, p. 278-295, APR 15 2017.
• Citações Web of Science: 3

Resumo

W. Specht (1940) proved that two n x n complex matrices A and B are unitarily similar if and only if trace w(A, A{*}) = trace w(B, B{*}) for every word w(x, y) in two noncommuting variables. We extend his criterion and its generalizations by N.A. Wiegmann (1961) and N. Jing (2015) to an arbitrary system A consisting of complex or real inner product spaces and linear mappings among them. We represent such a system by the directed graph Q(A), whose vertices are inner product spaces and arrows are linear mappings. Denote by (Q) over tilde (A) the directed graph obtained by enlarging to Q(A) the adjoint linear mappings. We prove that a system A is transformed by isometries of its spaces to a system 13 if and only if the traces of all closed directed walks in Q(A) and Q(13) coincide. (C) 2017 Elsevier Inc. All rights reserved. (AU)

• Processo FAPESP: 14/09310-5 - Estruturas algébricas e suas representações
• Beneficiário: Vyacheslav Futorny
• Modalidade de apoio: Auxílio à Pesquisa - Temático
• Processo FAPESP: 15/05864-9 - Problemas de classificação em álgebra linear e teoria de sistemas
• Beneficiário: Vyacheslav Futorny
• Modalidade de apoio: Auxílio à Pesquisa - Pesquisador Visitante - Internacional