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.

Key words： Schur complement; convergence; general H-matrices

References

1 Alanelli M, Hadjidimos A. A new iterative criterion for H-Matrices. SIAM J Matrix Anal Appl, 2006, 29(1): 160−176

2 Berman A, Plemmons R J. Nonnegative Matrices in the Mathematical Sciences. New York: Academic, 1979

3 Bru R, Corral C, Gimenez I, Mas J. Classes of general H-matrices. Linear Algebra Appl, 2008, 429: 2358−2366

4 Bru R, Corral C, Gimenez I, Mas J. Schur complement of general H-matrices. Numer Linear Algebra Appl, 2009, 16(11−12): 935−947

5 Bru R, Gimenez I, Hadjidimos A. Is A ∈ C n , n a general H-matrices. Linear Algebra Appl, 2012, 436: 364−380

6 Carlson D, Markham T. Schur complements of diagonally dominant matrices. Czechoslovak Math J, 1979, 29(104): 246−251

7 Cvetković L, Kostić V, Kovačević M, Szulc T. Further results on H-matrices and their Schur complements. Linear Algebra Appl, 2008, 198: 506−510

8 Elman H, Silvester D, Wathen A. Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics, Numerical Mathematics and Scientific Computation. Oxford: Oxford University Press, 2005

9 Fiedler M, Ptak V. On matrices with nonpositive offdiagonal elements and positive principal minors. Czechoslovak Math J, 1962, 12(87): 382−400

10 Fiedler M, Ptak V. Diagonally dominant matrices. Czechoslovak Math J, 1967, 17(92): 420−433

11 Golub G H, Van Loan C F. Matrix Computations. 3rd ed. Baltimore: Johns Hopkins University Press, 1996

12 Johnson C R. Inverse M-matrices. Linear Algebra Appl, 1982, 47: 195−216

13 Kolotilina L Yu. Nonsingularity/singularity criteria for nonstrictly block diagonally dominant matrices. Linear Algebra Appl, 2003, 359: 133−159

14 Kress R. Numerical Analysis. New York: Springer, 1998

15 Lei T G, Woo C W, Liu J Z, Zhang F. On the Schur complements of diagonally dominant matrices. In: Proceedings of the SIAM Conference on Applied Linear Algebra. 2003

16 Liao X. The Stability Theory and Application of Dynamic System. Beijing: National Defence industry Press, 2000 (in Chinese)

17 Liu Jianzhou, Huang Yunqing. Some properties on Schur complements of H-matrix and diagonally dominant matrices. Linear Algebra Appl, 2004, 389: 365−380

18 Liu Jianzhou, Huang Yunqing, Zhang Fuzhen. The Schur complements of generalized doubly diagonally dominant matrices. Linear Algebra Appl, 2004, 378: 231−244

19 Liu Jianzhou, Li Jicheng, Huang Zhuohong, Kong Xu. Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices. Linear Algebra Appl, 2008, 428: 1009−1030

20 Liu Jianzhou, Zhang Fuzhen. Disc separation of the Schur complement of diagonally dominant matrices and determinantal bounds. SIAM J Matrix Anal Appl, 2005, 27(3): 665−674

21 Ostrowski A M. Über die determinanten mit uberwiegender hauptdiagonale. Comment Math Helv, 1937, 10: 69−96

22 Plemmons R J. M-matrix characterizations. I−Nonsingular M-matrices. Linear Algebra Appl, 1977, 18: 175−188

23 Polman B. Incomplete blockwise factorizations of (block) H-matrices. Linear Algebra Appl, 1987, 90: 119−132

24 Varga R S. Matrix Iterative Analysis. Englewoods Cliffs: Prentice Hall, 1962 (reprinted and updated, Berlin: Springer, 2000)

25 You Zhaoyong. Nonsingular M-matrices. Wuhan: Huazhong Institute of Technology, 1981 (in Chinese)

26 Zhang Cheng-yi, Li Yao-tang. Diagonal dominant matrices and the determining of H-matrices and M-matrices. Guangxi Sciences, 2005, 12(3): 1161−1164 (in Chinese)

27 Zhang Cheng-yi, Li Yao-tang, Chen Feng. On Schur complements of block diagonally dominant matrices. Linear Algebra Appl, 2006, 414: 533−546

28 Zhang Cheng-yi, Luo Shuanghua, Xu Chengxian, Jiang Hongying. Schur complements of generally diagonally dominant matrices and criterion for irreducibility of matrices. Electron J Linear Algebra, 2009, 18: 69−87

29 Zhang Cheng-yi, Luo Shuanghua, Xu Fengmin, Xu Chengxian. The eigenvalue distribution on Schur complement of nonstrictly diagonally dominant matrices and general H-matrices. Electron J Linear Algebra, 2009, 18: 801−820

30 Zhang Cheng-yi, Xu Chengxian, Li Yao-tang. The eigenvalue distribution on Schur complements of H-matrices. Linear Algebra Appl, 2007, 422: 250−264

31 Zhang Cheng-yi, Xu Chengxian, Li Yao-tang. Nonsingularity/singularity criteria for nonstrictly generalized diagonally dominant matrices. In: Jiang Er-Xiong, Wang Chuan-long, eds. Advances in Matrix Theory and its Applications, Vol II. Proceedings of the Eighth International Conference on Matrix Theory and Its Applications, Taiyuan, China, July 2008. 2008, 425−428

32 Zhang Fuzhen. The Schur Complement and Its Applications. New York: Springer, 2005