Geodesic
The singularity/nonsingularity problem for matrices satisfying diagonal dominance conditions in terms of directed graphs
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVII, Tome 309 (2004), pp. 40-83 
L. Yu. Kolotilina. The singularity/nonsingularity problem for matrices satisfying diagonal dominance conditions in terms of directed graphs. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVII, Tome 309 (2004), pp. 40-83. http://geodesic.mathdoc.fr/item/ZNSL_2004_309_a4/
@article{ZNSL_2004_309_a4,
     author = {L. Yu. Kolotilina},
     title = {The singularity/nonsingularity problem for matrices satisfying diagonal dominance conditions in terms of directed graphs},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {40--83},
     year = {2004},
     volume = {309},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_309_a4/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The paper considers the singularity/nonsingularity problem for matrices satisfying certain conditions of diagonal dominance. The conditions considered extend the classical diagonal dominance conditions and involve the directed graph of the matrix in question. Furthermore, in the case of the so-called mixed diagonal dominance, the corresponding conditions are allowed to involve both row and column sums for an arbitrary finite set of matrices diagonally conjugated to the original matrix. Conditions sufficient for the nonsingularity of quasi-irreducible matrices strictly diagonally dominant in certain senses are established, as well as necessary and sufficient conditions of singularity/nonsingularity for weakly diagonally dominant matrices in the irreducible case. The results obtained are used to describe inclusion regions for eigenvalues of arbitrary matrices. In particular, a direct extension of the Gerschgorin ($r=1$) and Ostrowski–Brauer ($r=2$) theorems to $r\geqslant3$ is presented.

[1] Yu. A. Alpin, “Granitsy dlya perronovskogo kornya neotritsatelnoi matritsy, uchityvayuschie svoistva ee grafa”, Mat. zametki, 58 (1995), 635–637 | MR

[2] F. R. Gantmakher, Teoriya matrits, Nauka, M., 1953 | MR

[3] L. Yu. Kolotilina, “O teoreme Brualdi”, Zap. nauchn. semin. POMI, 284, 2002, 48–63 | MR

[4] L. Yu. Kolotilina, “Otsenki i neravenstva dlya perronovskogo kornya neotritsatelnoi matritsy”, Zap. nauchn. semin. POMI, 284, 2002, 77–122 | MR | Zbl

[5] L. Yu. Kolotilina, “Otsenki i neravenstva dlya perronovskogo kornya neotritsatelnoi matritsy, II”, Zap. nauchn. semin. POMI, 296, 2003, 60–88 | MR | Zbl

[6] A. Brauer, “Limits for the characteristic roots of a matrix, II”, Duke Math. J., 14 (1947), 21–26 | DOI | MR | Zbl

[7] A. Brauer, “Limits for the characteristic roots of a matrix, IV”, Duke Math. J., 19 (1952), 75–91 | DOI | MR | Zbl

[8] R. Brualdi, “Matrices, eigenvalues, and directed graphs”, Linear and Multilinear Algebra, 11 (1982), 143–165 | DOI | MR | Zbl

[9] M. Fiedler, V. Pták, “Cyclic products and an inequality for determinants”, Czechoslovak Math. J., 19 (1969), 428–450 | MR

[10] F. Harary, Graph Theory, Addison-Wesley Publ Co., 1969 | MR | Zbl

[11] R. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985 | MR | Zbl

[12] L. Yu. Kolotilina, “Nonsingularity/singularity criteria for nonstrictly block diagonally dominant matrices”, Linear Algebra Appl., 359 (2003), 133–159 | DOI | MR | Zbl

[13] L. Yu. Kolotilina, “Generalizations of the Ostrowski–Brauer theorem”, Linear Algebra Appl., 364 (2003), 65–80 | DOI | MR | Zbl

[14] B. Li, M. J. Tsatsomeros, “Doubly diagonally dominant matrices”, Linear Algebra Appl., 261 (1997), 221–235 | DOI | MR | Zbl

[15] M. Marcus, H. Minc, A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Inc., Boston, 1964 | MR | Zbl

[16] A. Ostrowski, “Über die Determinanten mit überwiegender Hauptdiagonale”, Comment. Math. Helv., 10 (1937), 69–96 | DOI | MR

[17] A. Ostrowski, “Über das Nichtverschwinden einer Klasse von Determinanten und die Lokalisierung der charakteristischen Wurzeln von Matrizen”, Compositio Math., 9 (1951), 209–226 | MR | Zbl

[18] Z. Xian, Gu Dunhe, “A note on A. Brauer's theorem”, Linear Algebra Appl., 196 (1994), 163–174 | DOI | MR | Zbl