Equivalence classes of e-matrices and associated eigenvalue localization regions

This paper is a continuation of a series of papers on related topics by the authors. It is divided into two main parts. In the first part, the set of n × n e-matrices (real constant row-sum matrices) is partitioned into equivalence classes which are called e-similarity classes. The relationships between the spectra of different matrices belonging to the same equivalence class, as well as their left and generalized left eigenspaces, are discussed. In the second part, the results obtained in the first part are applied to improve the location of eigenvalues of e-matrices. Associated localization Gershgorin regions of the so-called second type are obtained. A number of examples are provided.