Strengthening of localization theorems of Geršgorin and Brauer

Substantial refinements and generalizations of the two known theorems are proposed. We show how to take into account the eigenvectors and other invariants for the more accurate spectrum localization. It permits us to obtain simple and adventegeous criteria of regularity of finite Markov chains and corresponding convergence rate estimations. Similar considerations and a something greater care in proving of Brauer's theorem give us essentially stronger inequalities than in original formulation. We propose new nondegeneracy tests and a new theorem of spectrum separation by components of the localization set.