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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 961568, 6 pages
http://dx.doi.org/10.1155/2013/961568
Research Article

The Optimization on Ranks and Inertias of a Quadratic Hermitian Matrix Function and Its Applications

Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 3 December 2012; Accepted 9 January 2013

Academic Editor: Yang Zhang

Copyright © 2013 Yirong Yao. 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. D. L. Chu, Y. S. Hung, and H. J. Woerdeman, “Inertia and rank characterizations of some matrix expressions,” SIAM Journal on Matrix Analysis and Applications, vol. 31, no. 3, pp. 1187–1226, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  2. Z. H. He and Q. W. Wang, “Solutions to optimization problems on ranks and inertias of a matrix function with applications,” Applied Mathematics and Computation, vol. 219, no. 6, pp. 2989–3001, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  3. Y. Liu and Y. Tian, “Max-min problems on the ranks and inertias of the matrix expressions A-BXC±(BXC)* with applications,” Journal of Optimization Theory and Applications, vol. 148, no. 3, pp. 593–622, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. G. Marsaglia and G. P. H. Styan, “Equalities and inequalities for ranks of matrices,” Linear and Multilinear Algebra, vol. 2, pp. 269–292, 1974/75. View at Publisher · View at Google Scholar · View at MathSciNet
  5. X. Zhang, Q. W. Wang, and X. Liu, “Inertias and ranks of some Hermitian matrix functions with applications,” Central European Journal of Mathematics, vol. 10, no. 1, pp. 329–351, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  6. Q. W. Wang, Y. Zhou, and Q. Zhang, “Ranks of the common solution to six quaternion matrix equations,” Acta Mathematicae Applicatae Sinica, vol. 27, no. 3, pp. 443–462, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  7. Q. Zhang and Q. W. Wang, “The (P,Q)-(skew)symmetric extremal rank solutions to a system of quaternion matrix equations,” Applied Mathematics and Computation, vol. 217, no. 22, pp. 9286–9296, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. D. Z. Lian, Q. W. Wang, and Y. Tang, “Extreme ranks of a partial banded block quaternion matrix expression subject to some matrix equations with applications,” Algebra Colloquium, vol. 18, no. 2, pp. 333–346, 2011. View at Zentralblatt MATH · View at MathSciNet
  9. H. S. Zhang and Q. W. Wang, “Ranks of submatrices in a general solution to a quaternion system with applications,” Bulletin of the Korean Mathematical Society, vol. 48, no. 5, pp. 969–990, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Q. W. Wang and J. Jiang, “Extreme ranks of (skew-)Hermitian solutions to a quaternion matrix equation,” Electronic Journal of Linear Algebra, vol. 20, pp. 552–573, 2010. View at Zentralblatt MATH · View at MathSciNet
  11. I. A. Khan, Q. W. Wang, and G. J. Song, “Minimal ranks of some quaternion matrix expressions with applications,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 2031–2040, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Q. Wang, S. Yu, and W. Xie, “Extreme ranks of real matrices in solution of the quaternion matrix equation AXB=C with applications,” Algebra Colloquium, vol. 17, no. 2, pp. 345–360, 2010. View at Zentralblatt MATH · View at MathSciNet
  13. Q. W. Wang, H. S. Zhang, and G. J. Song, “A new solvable condition for a pair of generalized Sylvester equations,” Electronic Journal of Linear Algebra, vol. 18, pp. 289–301, 2009. View at Zentralblatt MATH · View at MathSciNet
  14. Q. W. Wang, S. W. Yu, and Q. Zhang, “The real solutions to a system of quaternion matrix equations with applications,” Communications in Algebra, vol. 37, no. 6, pp. 2060–2079, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  15. 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 Zentralblatt MATH · View at MathSciNet
  16. Q. W. Wang, H. S. Zhang, and S. W. Yu, “On solutions to the quaternion matrix equation AXB+CYD=E,” Electronic Journal of Linear Algebra, vol. 17, pp. 343–358, 2008. View at Zentralblatt MATH · View at MathSciNet
  17. Q. W. Wang, G. J. Song, and X. Liu, “Maximal and minimal ranks of the common solution of some linear matrix equations over an arbitrary division ring with applications,” Algebra Colloquium, vol. 16, no. 2, pp. 293–308, 2009. View at Zentralblatt MATH · View at MathSciNet
  18. Q. W. Wang, G. J. Song, and C. Y. Lin, “Rank equalities related to the generalized inverse AT,S(2) with applications,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 370–382, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. Q. W. Wang, S. W. Yu, and C. Y. Lin, “Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications,” Applied Mathematics and Computation, vol. 195, no. 2, pp. 733–744, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. Q. W. Wang, G. J. Song, and C. Y. Lin, “Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1517–1532, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. Y. Tian, “Formulas for calculating the extremum ranks and inertias of a four-term quadratic matrix-valued function and their applications,” Linear Algebra and its Applications, vol. 437, no. 3, pp. 835–859, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. Y. Tian, “Solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method,” Nonlinear Analysis. Theory, Methods & Applications, vol. 75, no. 2, pp. 717–734, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Y. Tian, “Maximization and minimization of the rank and inertia of the Hermitian matrix expression ABX(BX)* with applications,” Linear Algebra and its Applications, vol. 434, no. 10, pp. 2109–2139, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. Q. W. Wang, X. Zhang, and Z. H. He, “On the Hermitian structures of the solution to a pair of matrix equations,” Linear and Multilinear Algebra, vol. 61, no. 1, pp. 73–90, 2013. View at Publisher · View at Google Scholar