General H-matrices and their Schur complements

The definitions of θ-ray pattern matrix and θ-ray matrix are firstly proposed to establish some new results on nonsingularity/singularity and convergence of general H-matrices. Then some conditions on the matrix A ∈ ℂ^n×n and nonempty α ⊂ 〈n〉 = {1, 2,..., n} are proposed such that A is an invertible H-matrix if A(α) and A/α are both invertible H-matrices. Furthermore, the important results on Schur complement for general H-matrices are presented to give the different necessary and sufficient conditions for the matrix A ∈ H_n^M and the subset α ⊂ 〈n〉 such that the Schur complement matrix A/α ∈ H_{n-{pipe}α{pipe}}^I or A/α ∈ H_{n-{pipe}α{pipe}}^M or A/α ∈ H_{n-{pipe}α{pipe}}^S.