Bounds for the determinants and inverses of certain $H$-matrices

The paper studies a subclass, referred to as $PBDD(n_1,n_2)$, of the class of nonsingular $H$-matrices. A new characterization of matrices in $PBDD(n_1,n_2)$ is suggested. Two-sided bounds for the determinants of matrices in the class $PBDD(n_1,n_2)$ are derived, and their applications to strictly diagonally dominant matrices and to matrices with the Ostrowski–Brauer diagonal dominance are presented. An upper bound for the infinity norms of the inverses of matrices in $PBDD(n_1,n_2)$ is considered. Extensions to the case of block $k\times k$ matrices, $k\ge 2$, are addressed.