General H-matrices and their Schur complements

Abstract

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\in {\mathbb{C}}^{\mathit{n}\times \mathit{n}} and nonempty \alpha \subset 〈\mathit{n}〉=\{\mathrm{1},\mathrm{2},\cdots ,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\in H_n^M and the subset \alpha \subset 〈\mathit{n}〉 such that the Schur complement matrix A/\alpha \in H_{n-\left|\alpha \right|}^I\hspace{0.17em}\mathrm{o}\mathrm{r}\hspace{0.17em}A/\alpha \in H_{n-\left|a\right|}^M\hspace{0.17em}\mathrm{o}\mathrm{r}\hspace{0.17em}\mathit{A}/\alpha \in H_{n-\left|\alpha \right|}^S.

Keywords

Schur complement / convergence / general H-matrices

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg