A block generalization of Nekrasov matrices

The paper introduces the so-called generalized Nekrasov (GN) matrices, which provide a block extension of the conventional Nekrasov matrices. Basic properties of GN matrices are studied. In particular, it is proved that the GN matrices form a subclass of nonsingular $\mathcal{H}$-matrices and this subclass is closed with respect to Schur complements obtained by eliminating leading principal block submatrices. Also an upper bound for the $l_\infty$-norm of the inverse to a GN matrix is obtained, which generalizes the known bound for Nekrasov matrices. The case of block two-by-two GN matrices with scalar first diagonal block, which prove to be Dashnic–Zusmanovich matrices of the first type, is considered separately. The bounds obtained are applied to SDD matrices.