The class of generalized α-matrices is presented by Cvetkovi？, L. (2006), and
proved to be a subclass of H-matrices. In this paper, we present a new class of
matrices-generalized irreducible α-matrices, and prove that a generalized irreducible
α-matrix is an H-matrix. Furthermore, using the generalized arithmetic-geometric mean inequality, we obtain two new classes of H-matrices.
As applications of the obtained results, three regions including all the eigenvalues
of a matrix are given.
References
[1]
Bru, R., Cvetković, L., Kostić, V. and Pedroche, F. (2010) Characterization of α1-and α2-Matrices. Central European Journal of Mathematics, 8, 32-40.
[2]
Bru, R., Giménez, I. and Hadjidimos, A. (2012) Is a General H-Matrix? Linear Algebra and its Applications, 436, 364-380. https://doi.org/10.1016/j.laa.2011.03.009
[3]
Cvetković, L., Kostić, V. and Varga, R.S. (2004) A New Geršgorin-Type Eigenvalue Inclusion Set. Electronic Transactions on Numerical Analysis, 18, 73-80.
[4]
Cvetković, L. (2006) H-Matrix Theory vs. Eigenvalue Localization. Numerical Algorithms, 42, 229-245. https://doi.org/10.1007/s11075-006-9029-3
[5]
Cvetković, L., Kostić, V., Bru, R. and Pedroche, F. (2011) A Simple Generalization of Geršgorin’s Theorem. Advances in Computational Mathematics, 35, 271-280. https://doi.org/10.1007/s10444-009-9143-6
[6]
Li, C.Q. and Li, Y.T. (2011) Generalizations of Brauer’s Eigenvalue Localization Theorem. Electronic Journal of Linear Algebra, 22, 1168-1178. https://doi.org/10.13001/1081-3810.1500
[7]
Varga, R.S. and Krautstengl, A. (1999) On Geršgorin-Type Problems and Ovals of Cassini. Electronic Transactions on Numerical Analysis, 8, 15-20.
[8]
Huang, T.Z. (1995) A Note on Generalized Diagonally Dominant Matrices. Linear Algebra and its Applications, 225, 237-242. https://doi.org/10.1016/0024-3795(93)00368-A
[9]
Brauer, A. (1947) Limits for the Characteristic Roots of a Matrix II. Duke Mathematical Journal, 14, 21-26. https://doi.org/10.1215/S0012-7094-47-01403-8
[10]
Minkowski, H. (1900) Zur Theorieder Einheitenin den algebraischen Zahlkörpern. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1900, 90-93.
[11]
Lévy, L. (1881) Sur le possibilité du léquibreélectrique. Comptes Rendus Mathematique Academie des Sciences, Paris, 93, 706-708.
[12]
Geršgorin, S. (1931) Über die Abgrenzung der Eigenwerte einer Matrix. Izvestiya RAN. Seriya Matematicheskaya, 7, 749-754.
[13]
Desplanques, J. (1887) Théoèm dálgébre. J. de Math. Spec, 9, 12-13.
[14]
Taussky, O. (1948) Bounds for Characteristic Roots of Matrices. Duke Mathematical Journal, 15, 1043-1044. https://doi.org/10.1215/S0012-7094-48-01593-2
[15]
Taussky, O. (1949) A Recurring Theorem on Determinants. The American Mathematical Monthly, 56, 672-676. https://doi.org/10.2307/2305561
[16]
Varga, R.S. (2001) Geršgor-in-Type Eigenvalue Inclusion Theorems and Their Sharpness. Electronic Transactions on Numerical Analysis, 12, 113-133.
[17]
Varga, R.S. (2004) Geršgorin and His Circles. Springer-Verlag, Berlin. https://doi.org/10.1007/978-3-642-17798-9
[18]
Ostrowski, A. (1951) über das Nichverschwinder einer Klasse von Determinanten und die Lokalisierung der charakterischen Wurzeln von Matrizen. Compositio. Mathematica, 9, 209-226.
[19]
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
