Li, Bishan; Tsatsomeros, M. J. Doubly diagonally dominant matrices. (English) Zbl 0886.15027 Linear Algebra Appl. 261, 221-235 (1997). Doubly diagonally dominant matrices are those whose ovals of Cassini do not include the origin. This paper gives necessary and sufficient conditions for irreducible such matrices to be singular or an \(H\)-matrix. It is also shown that Schur complements of such matrices remain doubly diagonal dominant. Reviewer: F.Uhlig (Auburn) Cited in 2 ReviewsCited in 39 Documents MSC: 15B57 Hermitian, skew-Hermitian, and related matrices 15A09 Theory of matrix inversion and generalized inverses 15A45 Miscellaneous inequalities involving matrices 15B48 Positive matrices and their generalizations; cones of matrices Keywords:doubly diagonally dominant matrix; singular matrix; \(H\)-matrix; Schur complement; Cassini oval; irreducible matrix PDF BibTeX XML Cite \textit{B. Li} and \textit{M. J. Tsatsomeros}, Linear Algebra Appl. 261, 221--235 (1997; Zbl 0886.15027) Full Text: DOI OpenURL References: [1] Berman, A.; Plemmons, R., Nonnegative Matrices in the Mathematical Sciences, Classics Appl. Math. SIAM (1994) · Zbl 0815.15016 [2] Richard, A., Brualdi, Matrices, eigenvalues, and directed graphs, Linear and Multilinear Algebra, 11, 143-165 (1982) · Zbl 0484.15007 [3] Miroslav Fiedler, Special Matrices and Their Applications in Numerical Mathematics (1986), Martinus Nijhoff · Zbl 0677.65019 [4] Horn, Roger A.; Johnson, Charles R., Matrix Analysis (1990), Cambridge UP · Zbl 0704.15002 [5] Horn, Roger A.; Johnson, Charles R., Topics in Matrix Analysis (1991), Cambridge UP · Zbl 0729.15001 [6] Pang, Ming-Xian, A generalization of diagonal dominance of matrices and its applications, Acta Math. Appl. Sinica, 12, 1, 35-43 (1989) · Zbl 0663.15006 [8] Taussky, O., Bounds for characteristic roots of matrices, Duke Math. J., 15, 1043-1044 (1948) · Zbl 0031.24405 [9] Taussky, O., A recurring theorem on determinants, Amer. Math. Monthly, 10, 672-676 (1949) · Zbl 0036.01301 [10] Varga, R., On recurring theorems on diagonal dominance, Linear Algebra Appl., 13, 1-9 (1976) · Zbl 0336.15007 [11] Zhang, Xian; Gu, Dunhe, A note on A. Brauer’s theorem, Linear Algebra Appl., 196, 163-174 (1994) · Zbl 0801.15016 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.