The paper generalizes certain inclusion sets for the singular values of a square matrix to the case of an m × n matrix. In particular, it is shown that under a nonrestrictive assumption on the ordering of the matrix columns (if m < n) or the matrix rows (if m > n), a natural counterpart of the Gerschgorin theorem on the eigenvalue location is valid. Bibliography: 14 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press (1991).
C. R. Johnson, “A Gersgorin-type lower bound for the smallest singular value,” Linear Algebra Appl., 112, 1–7 (1989).
C. R. Johnson and T. Szulc, “Further lower bounds for the smallest singular value,” Linear Algebra Appl., 272, 169–179 (1998).
L. Yu. Kolotilina, “Bounds for the singular values of a matrix involving its sparsity pattern,” Zap. Nauchn. Semin. POMI, 323, 57–68 (2005).
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 circuit inclusion sets for the singular values of a square matrix,” Zap. Nauchn. Semin. POMI, 359, 78–93 (2008).
H.-B. Li, T.-Z. Huang, and H. Li, “Inclusion sets for singular values,” Linear Algebra Appl., 428, 2220–2235 (2008).
H.-B. Li, T.-Z. Huang, H. Li, and S.-Q. Shen, “Optimal Gerschgorin-type inclusion intervals of singular values,” Numer. Linear Algebra Appl., 14, 115–128 (2007).
J. S. Li, K. Yang, and Q.-X. Wang, “Digraphs and inclusion intervals of Brualdi-type for singular values,” Acta Math. Appl. Sin., 18, 471–476 (2002).
L. Li, “The undirected graph and estimates of matrix singular values,” Linear Algebra Appl., 285, 181–188 (1998).
L. L. Li, “Estimation for matrix singular values,” Comput. Math. Appl., 37, 9–15 (1999).
W. Li and Q. Chang, “Inclusion intervals of singular values and applications,” Comput. Math. Appl., 45, 1637–1646 (2003).
M. Marcus and H. Mine, A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Inc., Boston (1964).
L. Qi, “Some simple estimates for singular values of a matrix,” Linear Algebra Appl., 56, 105–119 (1984).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 94–105.
Rights and permissions
About this article
Cite this article
Kolotilina, L.Y. Inclusion sets for the singular values of a rectangular matrix. J Math Sci 157, 723–729 (2009). https://doi.org/10.1007/s10958-009-9355-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-009-9355-9