A note on the inclusion sets for singular values

Jun He; Yan-Min Liu; Jun-Kang Tian; Ze-Rong Ren

doi:10.3934/Math.2017.2.315

AIMS Mathematics, 2017, 2(2): 315-321. doi: 10.3934/Math.2017.2.315. Research article

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A note on the inclusion sets for singular values

Jun He, Yan-Min Liu, Jun-Kang Tian, Ze-Rong Ren

School of mathematics, Zunyi Normal College, Zunyi, Guizhou, 563006, P.R. China

Received: , Accepted: , Published:

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In this paper, for a given matrix $A=(a_{ij}) \in \mathbb{C}^{n\times n}$, in terms of $r_i$ and $c_i$, where $ r_i = \sum\limits_{j = 1,j \ne i}^n {\left| {a_{ij} } \right|}, \ \ c_i = \sum\limits_{j = 1,j \ne i}^n {\left| {a_{ji} } \right|} $, some new inclusion sets for singular values of a matrix are established. It is proved that the new inclusion sets are tighter than the Geršgorin-type sets [1] and the Brauer-type sets [2]. A numerical experiment show the efficiency of our new results.

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Keywords: Singular value; Matrix; Inclusion sets

Citation: Jun He, Yan-Min Liu, Jun-Kang Tian, Ze-Rong Ren. A note on the inclusion sets for singular values. AIMS Mathematics, 2017, 2(2): 315-321. doi: 10.3934/Math.2017.2.315

References:

1. L. Qi, Some simple estimates of singular values of a matrix, Linear Algebra Appl. 56 (1984), 105-119.
2. L.L. Li, Estimation for matrix singular values, Comput. Math. Appl. 37 (1999), 9-15.
3. R.A. Brualdi, Matrices, eigenvalues, and directed graphs, Linear Mutilinear Algebra, 11 (1982), 143-165.
4. G. Golub, W. Kahan, Calculating the singular values and pseudo-inverse of a matrix, SIAM J. Numer. Anal. 2 (1965), 205-224.
5. R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
6. R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
7. C.R. Johnson, A Geršgorin-type lower bound for the smallest singular value, Linear Algebra Appl., 112 (1989), 1-7.
8. C.R. Johnson, T. Szulc, Further lower bounds for the smallest singular value, Linear Algebra Appl. 272 (1998), 169-179.
9. W. Li, Q. Chang, Inclusion intervals of singular values and applications, Comput. Math. Appl., 45 (2003), 1637-1646.
10. Hou-Biao Li, Ting-Zhu Huang, Hong Li, Inclusion sets for singular values, Linear Algebra Appl. 428 (2008), 2220-2235.
11. R.S. Varga, Geršgorin and his circles, Springer Series in Computational Mathematics, Springer-Verlag, 2004.

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This article has been cited by:

1. L. Yu. Kolotilina, A New Subclass of the Class of Nonsingular H $$ \mathcal{H} $$ -Matrices and Related Inclusion Sets for Eigenvalues and Singular Values, Journal of Mathematical Sciences, 2019, 10.1007/s10958-019-04398-4

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Copyright Info: 2017, Jun He, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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