Exclusion sets in Dashnic–Zusmanovich localization sets

A new eigenvalue localization set is given by excluding some proper subsets that do not contain any eigenvalues of matrices from Dashnic–Zusmanovich localization sets. As an application, a sufficient condition for non-singularity of matrices is obtained. In order to locate all eigenvalues of matrices precisely, another set including two positive integers s and k is presented. By adjusting the parameters s and k , one can locate all eigenvalues and judge the non-singularity of matrices accurately.