A note on the inclusion sets for singular values

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

doi:10.3934/Math.2017.2.315

AIMS Mathematics

2017, Volume 2, Issue 2: 315-321. doi: 10.3934/Math.2017.2.315

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Research article

A note on the inclusion sets for singular values

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

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.
- 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[J]. AIMS Mathematics, 2017, 2(2): 315-321. doi: 10.3934/Math.2017.2.315

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Abstract

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.

References

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