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

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, $n\ge 3$, 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.

KEYWORDS:

• Gershgorin discs
• diagonal similarity
• sign skew-symmetric matrix

AMS SUBJECT CLASSIFICATIONS:

• 15A18
• 15B35

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 rand returns pseudorandom values drawn from the standard uniform distribution on the open interval (0,1).

Additional information

Funding

This work was supported by NSF grant DMS #0751964 (first and third authors) and by Portuguese Funds through FCT (Fundação para a Ciência e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020 (second and last authors).