Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 615069, 11 pages
http://dx.doi.org/10.1155/2009/615069
Research Article

Inverse Eigenvalue Problem of Unitary Hessenberg Matrices

1School of Mathematical Sciences, Guizhou Normal University, Xiamen 361005, China
2School of Mathematics and Computer Science, Xiamen University, Guiyang 550001, China

Received 19 June 2009; Revised 28 August 2009; Accepted 31 August 2009

Academic Editor: Binggen Zhang

Copyright © 2009 Chunhong Wu and Linzhang Lu. 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. L. R. Fletcher, “An inverse eigenvalue problem from control theory,” in Numerical Treatment of Inverse Problems in Differential and Integral Equations (Heidelberg, 1982), P. Deuflhard and E. Hairer, Eds., vol. 2 of Progress in Scientific Computing, pp. 161–170, Birkhäuser, Boston, Mass, USA, 1983. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. S. Zhou and H. Dai, The Algebraic Inverse Eigenvalue Problems, Henan Science and Technology Press, Zhenzhou, China, 1991.
  3. W. M. Wonham, Linear Multivariable Control: A Geometric Approach, vol. 10 of Applications of Mathematics, Springer, New York, NY, USA, 2nd edition, 1979. View at MathSciNet
  4. G. M. L. Gladwell, Inverse Problems in Vibration, vol. 9 of Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics. Dynamical Systems, Martinus Nijhoff, Dordrecht, The Netherlands, 1986. View at MathSciNet
  5. G. M. L. Gladwell, “The inverse problem for the vibrating beam,” Proceedings of the Royal Society, vol. 393, no. 1805, pp. 277–295, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. V. Barcilon, “Sufficient conditions for the solution of the inverse problem for a vibrating beam,” Inverse Problems, vol. 3, no. 2, pp. 181–193, 1987. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. K. T. Joseph, “Inverse eigenvalue problem in structural design,” AIAA Journal, vol. 30, no. 12, pp. 2890–2896, 1992. View at Google Scholar · View at Scopus
  8. N. Li, “A matrix inverse eigenvalue problem and its application,” Linear Algebra and Its Applications, vol. 266, pp. 143–152, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. C. T. Byrnes, “Pole placement by output feedback,” in Three Decads of Mathmaticla Systems Theory, vol. 135 of Lecture Notes in Control and Information Sciences, pp. 31–78, Springer, New York, NY, USA, 1989. View at Google Scholar
  10. J. Kautsky, N. K. Nichols, and P. Van Dooren, “Robust pole assignment in linear state feedback,” International Journal of Control, vol. 41, no. 5, pp. 1129–1155, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. K. E. Chu and N. Li, “Designing the Hopfield neural network via pole assignment,” International Journal of Systems Science, vol. 25, no. 4, pp. 669–681, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Q. Wu, Determination of the size of an object and its location in a cavity by eigenfrequency shifts, Ph.D. thesis, University of Sydney, Sydney, Australia, 1990.
  13. G. M. L. Gladwell and A. Morassi, “Estimating damage in a rod from change in nod position,” Inverse Problems in Engineering, vol. 7, pp. 215–233, 1999. View at Google Scholar
  14. G. M. L. Gladwell, “Inverse problems in vibration II,” Applied Mechanics Reviews, vol. 49, pp. 2–27, 1996. View at Google Scholar
  15. X. Chen and M. T. Chu, “On the least squares solution of inverse eigenvalue problems,” SIAM Journal on Numerical Analysis, vol. 33, no. 6, pp. 2417–2430, 1996. View at Google Scholar · View at Scopus
  16. L. Starek and D. J. Inman, “Symmetric inverse eigenvalue vibration problem and its application,” Mechanical Systems and Signal Processing, vol. 15, no. 1, pp. 11–29, 2001. View at Publisher · View at Google Scholar · View at Scopus
  17. U. Helmke and J. B. Moore, Optimization and Dynamical Systems, Communications and Control Engineering Series, Springer, London, UK, 1994. View at MathSciNet
  18. A. Kress and D. J. Inman, “Eigenstructure assignment using inverse eigenvalue methods,” Journal of Guidance, Control, and Dynamics, vol. 18, pp. 625–627, 1995. View at Google Scholar
  19. S. T. Smith, Geometric optimation methods for adaptive filtering, Ph.D. thesis, Harvard University, Cambridge, Mass, USA, 1993.
  20. S. J. Wang and S. Y. Chu, “An algebraic approach to the inverse eigenvalue problem for a quantum system with a dynamical group,” Journal of Physics A, vol. 27, no. 16, pp. 5655–5671, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. Q. Wu, “An inverse eigenvalue problem of symmetric multilayered media,” Applied Acoustics, vol. 45, no. 1, pp. 61–80, 1995. View at Google Scholar · View at Scopus
  22. M. Yamamoto, “Inverse eigenvalue problem for a vibration of a string with viscous drag,” Journal of Mathematical Analysis and Applications, vol. 152, no. 1, pp. 20–34, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. Baruch, “Optimation procedure to correct stiffness and flexibility matrices using vibration test,” AIAA Journal, vol. 16, no. 11, pp. 1208–1210, 1978. View at Google Scholar · View at Scopus
  24. G. S. Ammar and Ch. Y. He, “On an inverse eigenvalue problem for unitary Hessenberg matrices,” Linear Algebra and Its Applications, vol. 218, pp. 263–271, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. W. B. Gragg, “Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle,” Journal of Computational and Applied Mathematics, vol. 46, no. 1-2, pp. 183–198, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. G. S. Ammar, W. B. Gragg, and L. Reichel, “Determination of Pisarenko frequency etimates as eigenvalues of an orthononal matrix,” in Advanced Algorithm and Architrchtures for Signal Processing II, vol. 826 of Proceedings of SPIE, pp. 143–145, 1987.
  27. L. Reichel and G. S. Ammar, “Fast approximation of dominant harmonics by solving an orthononal eigenvalue problem,” in Proceedings of the 2nd IMA Conference on Mathematices in Signal Processing, J. McWriter et al., Ed., pp. 575–591, University Press, Oxford, UK, 1990.
  28. G. S. Ammar, W. B. Gragg, and L. Reichel, “Constructing a unitary Hessenberg matrix from spectral data,” in Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms, G. H. Golub and P. Van Dooren, Eds., pp. 385–396, Springer, New York, NY, USA, 1991. View at Google Scholar
  29. L. Reichel, G. S. Ammar, and W. B. Gragg, “Discrete least squares approximation by trigonometric polynomials,” Mathematics of Computation, vol. 57, no. 195, pp. 273–289, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. P. Delsarte and Y. Genin, “Tridiagonal approach to the algebraic environment of Toeplitz matrices. I. Basic results,” SIAM Journal on Matrix Analysis and Applications, vol. 12, no. 2, pp. 220–238, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. P. Delsarte and Y. Genin, “Tridiagonal approach to the algebraic environment of Toeplitz matrices. II. Zero and eigenvalue problems,” SIAM Journal on Matrix Analysis and Applications, vol. 12, no. 3, pp. 432–448, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. A. Bunse-Gerstner and C. Y. He, “On a Sturm sequence of polynomials for unitary Hessenberg matrices,” SIAM Journal on Matrix Analysis and Applications, vol. 16, no. 4, pp. 1043–1055, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet