Congruences and canonical forms for a positive matrix: Application to the Schweinler-Wigner extremum principle

Journal of Mathematical Physics

Volumn 40, Issue 7, 1999, Pages 3632-3642

  Simon, R ^a   Chaturvedi, S ^b   Srinivasan, V ^b  

^a INSTITUTE OF MATHEMATICAL SCIENCES   (India)

^b UNIVERSITY OF HYDERABAD   (India)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033243844     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532913     Document Type: Article

References (20)

1. See, for instance, F. C. Gantmacher, The Theory of Matrices (Chelsea, New York, 1960), Vol. 1.
2. J. Williamson, Am. J. Math. 58, 141 (1936); 59, 599 (1936); 61, 897 (1936). Williamson's results are more general than the theorem quoted, and obtain all the different canonical forms a real symmetric (not necessarily positive definite) matrix can take under congruence by the real symplectic group. The results of Williamson are summarized in a manner that should appeal to physicists, in V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978), Appendix 6.
3. J. Williamson, Am. J. Math. 58, 141 (1936); 59, 599 (1936); 61, 897 (1936). Williamson's results are more general than the theorem quoted, and obtain all the different canonical forms a real symmetric (not necessarily positive definite) matrix can take under congruence by the real symplectic group. The results of Williamson are summarized in a manner that should appeal to physicists, in V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978), Appendix 6.
4. J. Williamson, Am. J. Math. 58, 141 (1936); 59, 599 (1936); 61, 897 (1936). Williamson's results are more general than the theorem quoted, and obtain all the different canonical forms a real symmetric (not necessarily positive definite) matrix can take under congruence by the real symplectic group. The results of Williamson are summarized in a manner that should appeal to physicists, in V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978), Appendix 6.
5. J. Williamson, Am. J. Math. 58, 141 (1936); 59, 599 (1936); 61, 897 (1936). Williamson's results are more general than the theorem quoted, and obtain all the different canonical forms a real symmetric (not necessarily positive definite) matrix can take under congruence by the real symplectic group. The results of Williamson are summarized in a manner that should appeal to physicists, in V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978), Appendix 6.
6. J. Moser, Commun. Pure Appl. Math. 11, 81 (1958); A. Weinstein, Bull. Am. Math. Soc. 75, 814 (1971); N. Burgoyne and R. Cushman, Celest. Mech. 8, 435 (1974); J. Laub and K. Meyer, ibid. 9, 213 (1974).
7. J. Moser, Commun. Pure Appl. Math. 11, 81 (1958); A. Weinstein, Bull. Am. Math. Soc. 75, 814 (1971); N. Burgoyne and R. Cushman, Celest. Mech. 8, 435 (1974); J. Laub and K. Meyer, ibid. 9, 213 (1974).
8. J. Moser, Commun. Pure Appl. Math. 11, 81 (1958); A. Weinstein, Bull. Am. Math. Soc. 75, 814 (1971); N. Burgoyne and R. Cushman, Celest. Mech. 8, 435 (1974); J. Laub and K. Meyer, ibid. 9, 213 (1974).
9. J. Moser, Commun. Pure Appl. Math. 11, 81 (1958); A. Weinstein, Bull. Am. Math. Soc. 75, 814 (1971); N. Burgoyne and R. Cushman, Celest. Mech. 8, 435 (1974); J. Laub and K. Meyer, ibid. 9, 213 (1974).
10. A. J. Dragt, F. Neri, and G. Rangarajan, Phys. Rev. A 45, 2572 (1992); E. C. G. Sudarshan, C. B. Chiu, and G. Bhamathi, ibid. 52, 43 (1995).
11. A. J. Dragt, F. Neri, and G. Rangarajan, Phys. Rev. A 45, 2572 (1992); E. C. G. Sudarshan, C. B. Chiu, and G. Bhamathi, ibid. 52, 43 (1995).
12. R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 36, 3868 (1987); R. Simon, N. Mukunda, and B. Dutta, ibid. 49, 1567 (1994); Arvind, B. Dutta, N. Mukunda, and R. Simon, Pramana, J. Phys. 45, 471 (1995); Arvind, B. Dutta, N. Mukunda, and R. Simon, Phys. Rev. A 52, 1609 (1995).
13. R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 36, 3868 (1987); R. Simon, N. Mukunda, and B. Dutta, ibid. 49, 1567 (1994); Arvind, B. Dutta, N. Mukunda, and R. Simon, Pramana, J. Phys. 45, 471 (1995); Arvind, B. Dutta, N. Mukunda, and R. Simon, Phys. Rev. A 52, 1609 (1995).
14. R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 36, 3868 (1987); R. Simon, N. Mukunda, and B. Dutta, ibid. 49, 1567 (1994); Arvind, B. Dutta, N. Mukunda, and R. Simon, Pramana, J. Phys. 45, 471 (1995); Arvind, B. Dutta, N. Mukunda, and R. Simon, Phys. Rev. A 52, 1609 (1995).
15. R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 36, 3868 (1987); R. Simon, N. Mukunda, and B. Dutta, ibid. 49, 1567 (1994); Arvind, B. Dutta, N. Mukunda, and R. Simon, Pramana, J. Phys. 45, 471 (1995); Arvind, B. Dutta, N. Mukunda, and R. Simon, Phys. Rev. A 52, 1609 (1995).
16. H. C. Schweinler and E. P. Wigner, J. Math. Phys. 11, 1693 (1970).
17. See, for instance, C. K. Chui, Wavelet Analysis and its Applications (Academic, San Diego, 1992).
18. 2⊗K, with K diagonal, is the canonical form for a real antisymmetric matrix under rotation. Further K can be chosen to be non-negative, in general, and positive definite when the antisymmetric matrix is nonsingular.
19. S. Chaturvedi, A. K. Kapoor, and V. Srinivasan, J. Phys. A 31, L367 (1998).
20. R. Simon, N. Mukunda, and B. Dutta, Phys. Rev. A 49, 1567 (1994).