Publicação
Inseparable gersgorin discs and the existence of conjugate complex eigenvalues of real matrices
| Resumo: | 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. |
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| Autores principais: | Johnson, Charles |
| Outros Autores: | Zhang, Yulin; Qiu, Frank; Ferreira, Carla |
| Assunto: | Gershgorin discs diagonal similarity sign skew-symmetric matrix Ciências Naturais::Matemáticas |
| Ano: | 2023 |
| País: | Portugal |
| Tipo de documento: | artigo |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade do Minho |
| Idioma: | inglês |
| Origem: | RepositóriUM - Universidade do Minho |
| Resumo: | 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. |
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