Further block generalizations of Nekrasov matrices

The paper continues the study of block generalizations of Nekrasov matrices and introduces two new classes of the so-called $\widetilde{\mathrm{G}}\mathrm{N}$ and $\mathrm{BJN}$ matrices and compares them with the previously introduced class of $\mathrm{GN}$ matrices. Different properties of $\widetilde{\mathrm{G}}\mathrm{N}$ and $\mathrm{BJN}$ matrices are established. In particular, it is proved that the classes $\{\widetilde{\mathrm{G}}\mathrm{N}\}$ and $\{\mathrm{BJN}\}$ are closed with respect to Schur complements and monotone with respect to block partitioning. Also upper bounds for the norms of inverses $\|A^{-1}\|_\infty$ of $\mathrm{GN}$, $\widetilde{\mathrm{G}}\mathrm{N}$, and $\mathrm{BJN}$ matrices $A$ are considered. General results obtained are specialized to the case of block two-by-two matrices with scalar first diagonal block.