Reduced eigenvalue inclusion sets and related classes of nonsingular matrices

The paper considers the classes of nonsingular matrices, referred to as PSDD and PDZ matrices, that are associated with the Gerschgorin disks and Dashnic–Zusmanovich eigenvalue inclusion sets from which some subsets are excluded. It is demonstrated that PSDD and PDZ matrices are obtained by permuting rows of SDD and Dashnic–Zusmanovich (DZ) matrices, respectively. Based on these results, for PSDD and PDZ matrices $A$ upper bounds for the $l_\infty$-norm of the product $A^{-1}Q$ of the inverse matrix times a rectangular matrix $Q$ are derived.