Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices

In this paper, we prove that the diagonal-Schur complement of a strictly doubly diagonally dominant matrix is strictly doubly diagonally dominant matrix. The same holds for the diagonal-Schur complement of a strictly generalized doubly diagonally dominant matrix and a nonsingular H-matrix. We point out that under certain assumptions, the diagonal-Schur complement of a strictly doubly (doubly product) γ-diagonally dominant matrix is also strictly doubly (doubly product) γ-diagonally dominant. Further, we provide the distribution of the real parts of eigenvalues of a diagonal-Schur complement of H-matrix. We also show that the Schur complement of a γ-diagonally dominant matrix is not always γ-diagonally dominant by a numerical example, and then obtain a sufficient condition to ensure that the Schur complement of a γ-diagonally dominant matrix is γ-diagonally dominant.