Research article

A note on the inclusion sets for singular values

  • Received: 01 February 2017 Accepted: 15 May 2017 Published: 23 May 2017
  • 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.

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

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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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    [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.
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    [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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  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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