New inclusion sets for singular values

• Main Authors: Jun He, Yan-Min Liu, Jun-Kang Tian, Ze-Rong Ren
• Format: Article
• Language: English
• Published: SpringerOpen 2017-03-01
• Series: Journal of Inequalities and Applications
• Subjects: singular value; matrix; inclusion sets
• Online Access: http://link.springer.com/article/10.1186/s13660-017-1337-8

Abstract In this paper, for a given matrix A = ( a i j ) ∈ C n × n $A=(a_{ij}) \in\mathbb{C}^{n\times n}$ , in terms of r i $r_{i}$ and c i $c_{i}$ , where r i = ∑ j = 1 , j ≠ i n | a i j | $r_{i} = \sum _{j = 1,j \ne i}^{n} {\vert {a_{ij} } \vert }$ , c i = ∑ j = 1 , j ≠ i n | a j i | $c_{i} = \sum _{j = 1,j \ne i}^{n} {\vert {a_{ji} } \vert }$ , some new inclusion sets for singular values of the matrix are established. It is proved that the new inclusion sets are tighter than the Geršgorin-type sets (Qi in Linear Algebra Appl. 56:105-119, 1984) and the Brauer-type sets (Li in Comput. Math. Appl. 37:9-15, 1999). A numerical experiment shows the efficiency of our new results.