The paper suggests generalizations of some known sufficient nonsingularity conditions for matrices with constant principal diagonal and the corresponding eigenvalue inclusion sets to the cases of arbitrary matrices and matrices with nonzero diagonal entries. Bibliography: 11 titles.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Brauer, “Limits for the characteristic roots of a matrix: II,” Duke Math. J., 14, 21–26 (1947).
L. Cvetković, V. Kostić, R. Bru, and F. Pedroche, “A simple generalization of Geršgorin’s theorem,” Adv. Comput. Math., 35, 271–280 (2011).
S. Gerschgorin, “Über die Abgrenzung der Eigenwerte einer Matrix,” Izv. Akad. Nauk SSSR, Ser. Fiz.-Mat., 6, 749–754 (1931).
L. Yu. Kolotilina, “Generalizations of the Ostrowski–Brauer theorem,” Linear Algebra Appl., 364, 65–80 (2003).
L. Yu. Kolotilina, “The singularity/nonsingularity problem for matrices satisfying diagonal dominance conditions in terms of directed graphs,” Zap. Nauchn. Semin. POMI, 309, 40–83 (2005).
L. Yu. Kolotilina, “Bounds for the Perron root, singularity/nonsingularity conditions, and eigenvalue inclusion sets,” Numer. Algor., 42, 247–280 (2006).
C. Q. Li and Y. T. Li, “New regions including eigenvalues of Toeplitz matrices,” Linear Multilinear Algebra, 62, 229–241 (2014).
C. Q. Li, W. Q. Zhang, and Y. T. Li, “A new eigenvalue inclusion set for matrices with a constant main diagonal,” to appear in JIA (2015).
A. Melman, “Ovals of Cassini for Toeplitz matrices,” Linear Multilinear Algebra, 60, 189–199 (2012).
A. Ostrowski, “Über die Determinanten mit überwiegender Hauptdiagonale,” Comment. Math. Helv., 10, 69–96 (1937).
R. S. Varga, Geršgorin and His Circles, Springer, Berlin (2004).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 439, 2015, pp. 128–144.
Translated by L. Yu. Kolotilina.
Rights and permissions
About this article
Cite this article
Kolotilina, L.Y. New Nonsingularity Conditions for General Matrices and the Associated Eigenvalue Inclusion Sets. J Math Sci 216, 805–815 (2016). https://doi.org/10.1007/s10958-016-2946-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-016-2946-3