1. Inseparable Gershgorin discs and the existence of conjugate complex eigenvalues of real matrices.
- Author
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Johnson, Charles, Zhang, Yulin, Qiu, Frank, and Ferreira, Carla
- Subjects
EIGENVALUES ,MATRICES (Mathematics) - Abstract
We investigate the converse of the known fact that if the Gershgorin 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\ge 3 $ 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. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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