As I understood all complex eigenvalue are coming in complex conjugate pairs. Additionally if all the circle don't overlap so in each circle only one eigenvalue exist and if all the circle's centres are on real axis so not way that complex eigenvalue exit. Am I right?
In addition I would like to know if their is any other information that I can learn about existence of complex eigenvalue from Gershgorin circle?
If the matrix has real entries, or at least real entries in the main diagonal, all Gershgorin circles intersect the real line. Therefore, the circle theorem cannot establish the existence of complex eigenvalues.