Topics on a generalization of Gershgorinʹs theorem

We construct two classes of 3 × 3 and 4 × 4 real symmetric matrices, and establish sufficient conditions for the spectrum of a matrix A in each class to be disjoint from its kth order Gershgorin region. This provides a partial answer to a question raised by Newman and Thompson. The problem of providing sufficient conditions for the localization of the spectrum of a matrix in its kth order Gershgorin region is also discussed.