We introduce a new subclass of $H$-matrices called partitioned Dashnic-Zusmanovich type (DZT) matrices and present the corresponding scaling matrices for this kind of matrices. There are three major applications. The first application is to provide equivalent eigenvalue localization related to index partition by using the nonsingularity of the new subclass. By taking some specific partitions, we provide other forms of eigenvalue localization sets that generalize and improve some well-known eigenvalue localization sets. The second application is to obtain an upper bound on the infinite norm of the inverse of partitioned DZT matrices using scaling matrices. The third application is to give an error bound of the linear complementarity problems (LCPs) by using scaling matrices. Additionally, we give another upper bound of the infinite norm and error bound of the LCPs by a reduction method, which transforms the given partitioned DZT matrix into the corresponding DZT matrix by partition and summation. The results obtained by the reduction method are generalizations of some known conclusions.