The paper presents new nonsingularity conditions for n ×n matrices, which involve a subset S of the index set {1, . . ., n} and take into consideration the matrix sparsity pattern. It is shown that the matrices satisfying these conditions form a subclass of the class of nonsingular \( \mathcal{H} \)-matrices, which contains some known matrix classes such as the class of doubly strictly diagonally dominant (DSDD) matrices and the class of Dashnic–Zusmanovich type (DZT) matrices. The nonsingularity conditions established are used to obtain the corresponding eigenvalue inclusion sets, which, in their turn, are used in deriving new inclusion sets for the singular values of a square matrix, improving some recently suggested ones.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York etc. (1979).
A. Brauer, “Limits for the characteristic roots of a matrix: II,” Duke Math. J., 14, 21–26 (1947).
L. Cvetković, “H-matrix theory vs. eigenvalue localization,” Numer. Algorithms, 42, 229–245 (2007).
Jun He, Yan-Min Liu, Yun-Kang Tian, and Ze-Rong Ren, “A note on the inclusion set for singular values,” AIMS Mathematics, 2(2), 315–321 (2017).
Jun He, Yan-Min Liu, Yun-Kang Tian, and Ze-Rong Ren, “New inclusion sets for singular values,” J. Ineq. Appl., 64 (2017), DOI https://doi.org/10.1186/s13660-017-1337-8.
L. Yu. Kolotilina, “Generalizations of the Ostrowski–Brauer theorem,” Linear Algebra Appl., 364, 65–80 (2003).
L. Yu. Kolotilina, “Inclusion sets for the singular values of a square matrix,” Zap. Nauchn. Semin. POMI, 359, 52–77 (2008).
L. Yu. Kolotilina, “On Dashnic–Zusmanovich (DZ) and Dashnic–Zusmanovich type (DZT) matrices and their inverses,” Zap. Nauchn. Semin. POMI, 472, 145–165 (2018).
A. M. Ostrowski, “Über die Determinanten mit überwiegender Hauptdiagonale,” Comment. Math. Helv., 10, 69–96 (1937).
L. Qi, “Some simple estimates for singular values of a matrix,” Linear Algebra Appl., 56, 105–119 (1984).
Jianxing Zhao, Qilong Liu, Chaoqian Li, and Yaotang Li, “Dashnic–Zusmanovich type matrices: a new subclass of nonsingular H-matrices,” Linear Algebra Appl., 552, 277–287 (2018).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 472, 2018, pp. 166–178.
Rights and permissions
About this article
Cite this article
Kolotilina, L.Y. A New Subclass of the Class of Nonsingular \( \mathcal{H} \)-Matrices and Related Inclusion Sets for Eigenvalues and Singular Values. J Math Sci 240, 813–821 (2019). https://doi.org/10.1007/s10958-019-04398-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-019-04398-4