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On Classes of Matrices with Variants of the Diagonal Dominance Property
Department of Mathematics, Shanghai University, Shanghai, China
- 1 Department of Mathematics, Shanghai University, Shanghai, China
Advances in Linear Algebra & Matrix Theory·Volume 07 (2017)·Pages 37–65·Published 21 June 2017·DOI10.4236/alamt.2017.72005
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Abstract
We study the relations between several classes of matrices with variants of the diagonal dominance property, and identify those classes which form pairs of incomparable classes. For an incomparable pair ( X 1 , X 2 ) of classes of matrices with variants of the diagonal dominance property, we also study the problem of providing sufficient conditions for the matrices in X i to be in X j with {i,j}={1,2}. The article is a continuation of a series of articles on the topic and related topics by the author; see [1][2][3][4].
KeywordsDoubly Diagonally DominantGeneralized Diagonally Dominant(S<sub>1</sub>S<sub>2</sub>) Separation Induced Diagonally DominantRow-Column Diagonally Dominant with Index &alpha
- Farid, F.O. (1995) Criteria for Invertibility of Diagonally Dominant Matrices. Linear Algebra and Its Applications, 215, 63-93. https://doi.org/10.1016/0024-3795(93)00072-8
- Farid, F.O. (1998) Topics on a Generalization of Gershgorin’s Theorem. Linear Algebra and Its Applications, 268, 91-116. https://doi.org/10.1016/S0024-3795(97)00030-X
- Farid, F.O. (2005) lp-Diagonally Dominant Symmetric Operators. Positivity, 9, 97-114. https://doi.org/10.1007/s11117-003-5371-z
- Farid, F.O. (2011) Notes on Matrices with Diagonally Dominant Properties. Linear Algebra and Its Applications, 435, 2793-2812. https://doi.org/10.1016/j.laa.2011.04.045
- Desplanques, J. (1887) Théorèm d’algébre. J. de Math. Spec., 9, 12-13.
- Lévy, L. (1881) Sur le possibilité du l'equibre électrique. Comptes Rendus de l'Académie des Sciences, 93, 706-708.
- Brauer, A. (1947) Limits for the Characteristic Roots of Matrices II. Duke Mathematical Journal, 14, 21-26. https://doi.org/10.1215/S0012-7094-47-01403-8
- Brauer, A. (1952) Limits for the Characteristic Roots of Matrices IV. Duke Mathematical Journal, 19, 75-91. https://doi.org/10.1215/S0012-7094-52-01910-8
- Brualdi, R.A. (1982) Matrices, Eigenvalues, and Directed Graphs. Linear and Multilinear Algebra, 11, 143-165.
- Fan, K. (1958) Note on Circular Disks Containing the Eigenvalues of a Matrix. Duke Mathematical Journal, 25, 441-445. https://doi.org/10.1215/S0012-7094-58-02538-9
- Gan, T.-B. and Huang, T.-Z. (2003) Simple Criteria for Nonsingular H-Matrices. Linear Algebra and Its Applications, 374, 317-326. https://doi.org/10.1016/S0024-3795(03)00646-3
- Gao, Y.-M. and Wang, X.-H. (1992) Criteria for Generalized Diagonally Dominant Matrices and M-Matrices. Linear Algebra and Its Applications, 169, 257-268. https://doi.org/10.1016/0024-3795(92)90182-A
- Gao, Y.-M. and Wang, X.-H. (1996) Criteria for Generalized Diagonally Dominant Matrices and M-Matrices. II. Linear Algebra and Its Applications, 248, 339-353. https://doi.org/10.1016/0024-3795(95)00251-0
- Geršgorin, S. (1931) über die Abgrenzung der Eigenwerte einer Matrix. Izvestiya Akademii Nauk SSR, 7, 749-754.
- Hadjidimos, A. (2012) Irreducibility and Extensions of Ostrowski's Theorem. Linear Algebra and Its Applications, 436, 2156-2168. https://doi.org/10.1016/j.laa.2011.11.035