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, , 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.