Congruences and Canonical Forms for a Positive Matrix: Application to the Schweinler-Wigner Extremum Principle
Simon, R. ; Chaturvedi, S. ; Srinivasan, V.
arXiv, 9811003 / Harvested from arXiv
Text on arXiv
It is shown that a $N\times N$ real symmetric [complex hermitian] positive definite matrix $V$ is congruent to a diagonal matrix modulo a pseudo-orthogonal [pseudo-unitary] matrix in $SO(m,n)$ [ $SU(m,n)$], for any choice of partition $N=m+n$. It is further shown that the method of proof in this context can easily be adapted to obtain a rather simple proof of Williamson's theorem which states that if $N$ is even then $V$ is congruent also to a diagonal matrix modulo a symplectic matrix in $Sp(N,{\cal R})$ [$Sp(N,{\cal C})$]. Applications of these results considered include a generalization of the Schweinler-Wigner method of `orthogonalization based on an extremum principle' to construct pseudo-orthogonal and symplectic bases from a given set of linearly independent vectors.
Publié le : 1998-11-04
Classification:  Mathematical Physics,  Quantum Physics 
@article{9811003,
     author = {Simon, R. and Chaturvedi, S. and Srinivasan, V.},
     title = {Congruences and Canonical Forms for a Positive Matrix: Application to
  the Schweinler-Wigner Extremum Principle},
     journal = {arXiv},
     volume = {1998},
     number = {0},
     year = {1998},
     language = {en},
     url = {http://dml.mathdoc.fr/item/9811003}
}

Simon, R.; Chaturvedi, S.; Srinivasan, V. Congruences and Canonical Forms for a Positive Matrix: Application to
  the Schweinler-Wigner Extremum Principle. arXiv, Tome 1998 (1998) no. 0, . http://gdmltest.u-ga.fr/item/9811003/