A new subclass of the class of nonsingular $\mathcal H$-matrices and related inclusion sets for eigenvalues and singular values

The paper presents new nonsingularity conditions for $n\times n$ matrices, which involve a subset $S$ of the index set $\{1, \dots,n\}$ and take into consideration the matrix sparsity pattern. It is shown that the matrices satisfying these conditions form a subclass of the class of nonsingular $\mathcal H$-matrices, which contains some known matrix classes such as the class of doubly strictly diagonally dominant (DSDD) matrices and the class of Dashnic–Zusmanovich type (DZT) matrices.
The nonsingularity conditions established are used to obtain the corresponding eigenvalue inclusion sets, which, in their turn, are used in deriving new inclusion sets for the singular values of a square matrix, improving some recently suggested ones.