Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2013, Article ID 267035, 6 pages
http://dx.doi.org/10.1155/2013/267035
Research Article

Upper and Lower Bounds for Ranks of the Matrix Expression

School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, China

Received 2 September 2013; Accepted 18 December 2013

Academic Editor: Changbum Chun

Copyright © 2013 Zhiping Xiong. 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. R. Penrose, “A genaralized inverse for matrices,” Proceedings of the Cambridge Philosophical Society, vol. 52, pp. 406–413, 1955. View at Google Scholar
  2. A. Ben-Iserael and T. N. E. Greville, Generalized Inverses: Theorey and Applications, Wiley-Interscience, New York, NY, USA, 1974.
  3. A. Ben-Iserael and T. N. E. Greville, Generalized Inverses: Theorey and Applications, Springer, New York, NY, USA, 2nd edition, 2002.
  4. G. Wang, Y. Wei, and S. Qiao, Generalized Inverses: Theory and Computations, Science Press, Beijing, China, 2004.
  5. C. L. David, Linear Algebra and Its Applications, Addison-Wesley, Reading, Mass, USA, 1994.
  6. H. N. Braden, “The equations ATX±XTA=B,” SIAM Journal on Matrix Analysis and Applications, vol. 20, no. 2, pp. 295–302, 1999. View at Google Scholar · View at Scopus
  7. N. Cohan, C. R. Johnson, L. Rodman, and H. J. Woerdeman, “Ranks of completions of partial matrices,” Operator Theory, vol. 40, pp. 165–185, 1989. View at Google Scholar
  8. C. Davis, “Completing a matrix so as to minimize the rank,” Operator Theory, vol. 29, pp. 87–95, 1988. View at Google Scholar
  9. J. Gross, “Nonnegative-definite and positive-definite solutions to the matrix equation AXA*=B-revisited,” Linear Algebra and Its Applications, vol. 321, no. 1–3, pp. 123–129, 2000. View at Publisher · View at Google Scholar · View at Scopus
  10. N. I. M. Gould, M. E. Hribar, and J. Nocedal, “On the solution of equality constrained quadratic programming problems arising in optimization,” SIAM Journal on Scientific Computing, vol. 23, no. 4, pp. 1376–1395, 2002. View at Publisher · View at Google Scholar · View at Scopus
  11. C. R. Johnson, “Matrix completion problems: a survey. In matrix theory and applications,” Proceedings of Symposia in Applied Mathematics, vol. 40, pp. 171–197, 1990. View at Google Scholar
  12. C. R. Johnson and G. T. Whitney, “Minimum rank completions,” Linear and Multilinear Algebra, vol. 28, pp. 271–273, 1991. View at Google Scholar
  13. Y. Tian, “The maximal and minimal ranks of some expression of generalized inverses of matrices,” Southeast Asian Bulletin of Mathematics, vol. 25, pp. 745–755, 2002. View at Publisher · View at Google Scholar
  14. 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 Zentralblatt MATH · View at Scopus
  15. Y. Tian, “Solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method,” Nonlinear Analysis, Theory, Methods and Applications, vol. 75, no. 2, pp. 717–734, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. H. J. Woerdeman, “Minimal rank completions for block matrices,” Linear Algebra and Its Applications C, vol. 121, pp. 105–122, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  17. Z. Xiong and Y. Qin, “On the inverse of a special Schur complement,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7679–7684, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  18. X. Zhan, Q. W. Wang, and X. Liu, “Inertias and ranks of some Hermitian matrix functions with application,” Central European Journal of Mathematics, vol. 10, no. 1, pp. 329–351, 2012. View at Publisher · View at Google Scholar
  19. Y. Tian, “Upper and lower bounds for ranks of matrix expressions using generalized inverses,” Linear Algebra and Its Applications, vol. 355, no. 1–3, pp. 187–214, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  20. Y. Tian and S. Cheng, “The maximal and minimal ranks of A-BXC with applications,” New York Journal of Mathematics, vol. 9, pp. 345–362, 2003. View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  21. L. Mihályffy, “An alternative representation of the generalized inverse of partitioned matrices,” Linear Algebra and Its Applications, vol. 4, no. 1, pp. 95–100, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  22. G. Marsaglia and G. P. H. Styan, “Equalities and inequalities for ranks of matrices,” Linear Multilinear Algebra, vol. 2, no. 3, pp. 269–292, 1974. View at Publisher · View at Google Scholar