Inseparable gersgorin discs and the existence of conjugate complex eigenvalues of real matrices

• Autor(a) principal: Johnson, Charles
• Data de Publicação: 2023
• Outros Autores: Zhang, Yulin, Qiu, Frank, Ferreira, Carla
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: https://hdl.handle.net/1822/83882
• topic: Gershgorin discs; diagonal similarity; sign skew-symmetric matrix; Ciências Naturais::Matemáticas; Science & Technology

We investigate the converse of the known fact that if the Gersgorin discs of a real n-by-n matrix may be separated by positive diagonal similarity then the eigenvalues are real. In the 2-by-2 case, with appropriate signs for the off-diagonal entries, we find that the converse is correct, which raises several questions. First, in the 3-by-3 case, the converse is not generally correct, but, empirically it is frequently true. Then, in the n-by-n case, n >=3, we find that if all the 2-by-2 principal submatrices have inseparable discs (\strongly inseparable discs"), the full matrix must have a nontrivial pair of conjugate complex eigenvalues (i.e., cannot have all real eigenvalues). This hypothesis cannot generally be weakened.