About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 457298, 7 pages
http://dx.doi.org/10.1155/2014/457298
Research Article

On Extremal Ranks and Least Squares Solutions Subject to a Rank Restriction

Department of Mathematics and Computational Science, Huainan Normal University, Anhui 232038, China

Received 16 February 2014; Revised 11 June 2014; Accepted 22 June 2014; Published 8 July 2014

Academic Editor: Sofiya Ostrovska

Copyright © 2014 Hongxing Wang and Yeguo Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Springer, Berlin, Germany, 2nd edition, 2003. View at MathSciNet
  2. F. Uhlig, “On the matrix equation AX=B with applications to the generators of a controllability matrix,” Linear Algebra and its Applications, vol. 85, pp. 203–209, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  3. Y. Tian, “Ranks of solutions of the matrix equation AXB=C,” Linear and Multilinear Algebra, vol. 51, no. 2, pp. 111–125, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. R. Li and Y. Liu, “Ranks of Hermitian solutions of the matrix equation AX=B,” Far East Journal of Mathematical Sciences, vol. 26, no. 1, pp. 117–126, 2007. View at MathSciNet
  5. Y. Li, F. Zhang, W. Guo, and J. Zhao, “Solutions with special structure to the linear matrix equation AX=B,” Computers & Mathematics with Applications, vol. 61, no. 2, pp. 374–383, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. Y. H. Liu, “Ranks of solutions of the linear matrix equation AX+YB=C,” Computers & Mathematics with Applications, vol. 52, no. 6-7, pp. 861–872, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. Q.-W. Wang and C.-K. Li, “Ranks and the least-norm of the general solution to a system of quaternion matrix equations,” Linear Algebra and Its Applications, vol. 430, no. 5-6, pp. 1626–1640, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. Q. Wang and Z. H. He, “Solvability conditions and general solution for mixed Sylvester equations,” Automatica, vol. 49, no. 9, pp. 2713–2719, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. Y. H. Liu, “Ranks of least squares solutions of the matrix equation AXB=C,” Computers & Mathematics with Applications, vol. 55, no. 6, pp. 1270–1278, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. K. C. Sou and A. Rantzer, “On a generalized matrix approximation problem in the spectral norm,” Linear Algebra and Its Applications, vol. 436, no. 7, pp. 2331–2341, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. M. Wei and D. Shen, “Minimum rank solutions to the matrix approximation problems in the spectral norm,” SIAM Journal on Matrix Analysis and Applications, vol. 33, no. 3, pp. 940–957, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. X. F. Duan, Q. W. Wang, and J. F. Li, “On the low-rank approximation arising in the generalized Karhunen-Loeve transform,” Abstract and Applied Analysis, vol. 2013, Article ID 528281, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  13. H. Wang and J. Xu, “Some results on characterizations of matrix partial orderings,” Journal of Applied Mathematics, vol. 2014, Article ID 408457, 6 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  14. Y. Liu and Y. Tian, “Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA*=B with applications,” Journal of Applied Mathematics and Computing, vol. 32, no. 2, pp. 289–301, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. Y. Tian and Y. Liu, “Extremal ranks of some symmetric matrix expressions with applications,” SIAM Journal on Matrix Analysis and Applications, vol. 28, no. 3, pp. 890–905, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. H. Wang, “On least squares solutions subject to a rank restriction,” Linear and Multilinear Algebra, 2014. View at Publisher · View at Google Scholar
  17. M. Wei, “Perturbation theory for the Eckart-Young-Mirsky theorem and the constrained total least squares problem,” Linear Algebra and Its Applications, vol. 280, no. 1–3, pp. 267–287, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. H. Y. Zha, “The restricted singular value decomposition of matrix triplets,” SIAM Journal on Matrix Analysis and Applications, vol. 12, no. 1, pp. 172–194, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. M. Wei and Q. Wang, “On rank-constrained Hermitian nonnegative-definite least squares solutions to the matrix equation AXAH=B,” International Journal of Computer Mathematics, vol. 84, no. 6, pp. 945–952, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. G. Marsaglia and G. P. H. Styan, “Equalities and inequalities for ranks of matrices,” Linear and Multilinear Algebra, vol. 2, pp. 269–292, 1974. View at Publisher · View at Google Scholar · View at MathSciNet
  21. Y. Tian, “More on maximal and minimal ranks of Schur complements with applications,” Applied Mathematics and Computation, vol. 152, no. 3, pp. 675–692, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. C. Eckart and G. Young, “The approximation of one matrix by another of lower rank,” Psychometrika, vol. 1, no. 3, pp. 211–218, 1936. View at Publisher · View at Google Scholar · View at Scopus
  23. L. Mirsky, “Symmetric gauge functions and unitarily invariant norms,” The Quarterly Journal of Mathematics, vol. 11, pp. 50–59, 1960. View at Publisher · View at Google Scholar · View at MathSciNet
  24. G. H. Golub and C. F. van Loan, Matrix computations, vol. 3 of Johns Hopkins Series in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, Md, USA, 2nd edition, 1989. View at MathSciNet
  25. D. L. Chu, H. C. Chan, and D. W. C. Ho, “Regularization of singular systems by derivative and proportional output feedback,” SIAM Journal on Matrix Analysis and Applications, vol. 19, no. 1, pp. 21–38, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus